# The RHIC Beam Energy Scan Program: Results from the PHENIX Experiment

Jeffery T. Mitchell (for the PHENIX111A list of members of the PHENIX
Collaboration and acknowledgements can be found at the end of this
issue.Collaboration)  Brookhaven National Laboratory, Building 510C, P.O. Box
5000, Upton, NY 11973-5000 USA

###### Abstract

The PHENIX Experiment at RHIC has conducted a beam energy scan at several
collision energies in order to search for signatures of the QCD critical point
and the onset of deconfinement. PHENIX has conducted measurements of
transverse energy production, muliplicity fluctuations, and the skewness and
kurtosis of net charge distributions. The data analyzed to date show no
significant indications of the presence of the critical point.

††journal: Nuclear Physics A

##  1 Introduction

Recent lattice QCD calculations predict that there is a first order phase
transition from hadronic matter to a Quark-Gluon Plasma that ends in a
critical point, with a continuous phase transition on the other side of the
critical point. The Relativistic Heavy Ion Collider (RHIC) has conducted a
program to probe different regions of the QCD phase diagram in the vicinity of
the possible critical point with a beam energy scan. During 2010 and 2011,
RHIC provided Au+Au collisions to PHENIX at $\sqrt{s_{NN}}=$ 200 GeV, 62.4
GeV, 39 GeV, 27 GeV, 19.6 GeV, and 7.7 GeV. Analysis of the data concentrates
on two strategies: looking for signs of the onset of deconfinement by
comparing to results at the top RHIC energy, and searching for direct
signatures of a critical point. Results from PHENIX covering transverse energy
production, multipicity fluctuations, and the skewness and kurtosis of net
charge distributions will be discussed in this article.

##  2 Transverse Energy Production

PHENIX has measured transverse energy ($E_{T}$) production in Au+Au collisions
at the following collision energies: 200, 62.4, 39, 27, 19.6, and 7.7 GeV.
These observables are closely related to the geometry of the system and are
fundamental measurements necessary to understand the global properties of the
collision. This work extends the previous PHENIX measurements in 200, 130, and
19.6 GeV Au+Au collisions [1]. Total $E_{T}$ production results are summarized
in Figure 1, which shows the excitation function of the estimated value of the
Bjorken energy density [2] expressed as

$\epsilon_{BJ}=\frac{1}{A_{\perp}\tau}\frac{dE_{T}}{dy},$ (1)

where $\tau$ is the formation time and $A_{\perp}$ is the transverse overlap
area of the nuclei. The Bjorken energy density increases monotonically over
the range of the RHIC beam energy scan. Also shown is the estimate for 200 GeV
U+U collisions taken during the 2012 running period. Although $E_{T}$
production dramatically increases at LHC energies compared to RHIC energies,
the shape of the distributions as a function of the number of participants,
$N_{part}$, is independent of the collision energy. This is illustrated in
Figure 2, which shows an overlay of the distributions of $dE_{T}/d\eta$
normalized by the number of participant pairs for 7.7 GeV, 200 GeV, and 2.76
TeV Au+Au collisions. The 200 GeV and 7.7 GeV distributions have been scaled
up by a factors of 2.6 and 9.7, respectively. The shape of the distribution as
a function of $N_{part}$ appears to be driven by the collision geometry.

Figure 1: The estimated value of the Bjorken energy density, $\epsilon_{BJ}$,
multiplied by the formation time in central Au+Au collisions at mid-rapidity
as a function of $\sqrt{s_{NN}}$. The open circle represents the estimate for
200 GeV U+U collisions. Figure 2: $dE_{T}/d\eta$ normalized by the number of
participant pairs as a function $N_{part}$. Overlayed are the distributions
from 7.7 GeV, 200 GeV, and 2.76 TeV Au+Au collisions. The PHENIX data has been
scaled up to overlay the ALICE data [3].

##  3 Charged Particle Multiplicity Fluctuations

Near the QCD critical point, it is expected that fluctuations in the charged
particle multiplicity will increase [4]. PHENIX has extended the previous
analysis of multiplicity fluctuations in 200 and 62.4 GeV Au+Au collisions [5]
to 39 and 7.7 GeV Au+Au collisions. Charged particle multiplicity fluctuations
are measured using the scaled variance, $\omega_{ch}=\sigma_{ch}/\mu_{ch}$,
which is the standard deviation scaled by the mean of the distribution. The
scaled variance is corrected for contributions due to non-dynamic impact
parameter fluctuations using the method described in [5]. Figure 3 shows the
PHENIX results for central collisions compared to results from the NA49
Collaboration [6] as a function of $\sqrt{s_{NN}}$. There is no indication of
the presence of a critical point from the PHENIX results alone.

Figure 3: Charged particle multiplicity fluctuations in central Au+Au (PHENIX)
and Pb+Pb (NA49) collisions expressed in terms of the scaled variance as a
function of $\sqrt{s_{NN}}$. Circles represent the results from PHENIX.

##  4 Higher Moments of Net Charge Distributions

The shapes of the distributions of the event-by-event net charge are expected
to be sensitive to the presence of the critical point [7]. PHENIX has measured
the skewness ($S=\langle(N-\langle N\rangle)^{3}\rangle/\sigma^{3}$) and the
kurtosis ($\kappa=\langle(N-\langle N\rangle)^{4}\rangle/\sigma^{4}-3$) of net
charge distributions in Au+Au collisions at 200, 62.4, 39, and 7.7 GeV. These
values are expressed in terms that can be associated with the quark number
susceptibilities, $\chi$: $S\sigma\approx\chi^{(3)}/\chi^{(2)}$ and
$\kappa\sigma^{2}\approx\chi^{(4)}/\chi^{(2)}$ [8]. The skewness and kurtosis
for central collisions are shown in Figure 4 as a function of $\sqrt{s_{NN}}$.
The data are compared to URQMD and HIJING simulation results processed through
the PHENIX acceptance and detector response. There is no excess above the
simulation results observed in the data at these four collision energies.

Figure 4: The skewness multiplied by the standard deviation and the kurtosis
multiplied by the variance from net charge distributions from central Au+Au
collisions. The circles represent the data. The grey error bars represent the
systematic errors. Also shown are URQMD and HIJING simulation results
processed through the PHENIX acceptance. The increase in the kurtosis from
URQMD and HIJING may be due to an increase in resonance production at 200 GeV.
The blue curve is the result from the Hadron Resonance Gas model [8].

##  5 Summary

Presented here are some of the PHENIX results from the RHIC beam energy scan
program. From the analyses completed to date, there is no significant
indication of the presence of the QCD critical point. However, many analyses
from PHENIX, particularly at $\sqrt{s_{NN}}=$ 27 GeV and 19.6 GeV, will be
available soon.

## References

  * [1] S.S. Adler et al., Phys. Rev. C 71, 034908 (2005).
  * [2] J. D. Bjorken, Phys. Rev. D 27, 140 (1983).
  * [3] C. Loizides et al., arXiv:1106.6324v1 (2011).
  * [4] M. Stephanov et al, Phys. Rev. D 60, 114028 (1999).
  * [5] A. Adare et al, Phys. Rev. C 78, 044902 (2008).
  * [6] C. Alt et al, Phys. Rev. C 78, 034914 (2008).
  * [7] R. V. Gavai and S. Gupta, Phys. Lett. B 696, 459 (2011).
  * [8] F. Karsch and K. Redlich, Phys. Lett. B 695, 136 (2011).


# Tunable Photon blockade with single atom in a cavity under
electromagnetically induced transparency

Jing Tang  1 Yuangang Deng  1,* and Chaohong Lee1,2,3  1Guangdong Provincial
Key Laboratory of Quantum Metrology and Sensing $\&$ School of Physics and
Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, People’s
Republic of China
2State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-
Sen University (Guangzhou Campus), Guangzhou 510275, People’s Republic of
China
3Synergetic Innovation Center for Quantum Effects and Applications, Hunan
Normal University, Changsha 410081, People’s Republic of China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present an experimental proposal to achieve a strong photon blockade by
employing electromagnetically induced transparency (EIT) with single alkaline-
earth-metal atom trapped in an optical cavity. In the presence of optical
Stark shift, both second-order correlation function and cavity transmission
exhibit asymmetric structures between the red and blue sidebands of the
cavity. For a weak control field, the photon quantum statistics for the
coherent transparency window (i.e. atomic quasi-dark state resonance) are
insensitive to the Stark shift, which should also be immune to the spontaneous
emission of the excited state by taking advantage of the intrinsic dark-state
polariton of EIT. Interestingly, by exploiting the interplay between Stark
shift and control field, the strong photon blockade at atomic quasi-dark state
resonance has an optimal second-order correlation function $g^{(2)}(0)\sim
10^{-4}$ and a high cavity transmission simultaneously. The underlying
physical mechanism is ascribed to the Stark shift enhanced spectrum
anharmonicity and the EIT hosted strong nonlinearity with loss-insensitive
atomic quasi-dark state resonance, which is essentially different from the
conventional proposal with emerging Kerr nonlinearity in cavity-EIT. Our
results reveal a new strategy to realize high-quality single photon sources,
which could open up a new avenue for engineering nonclassical quantum states
in cavity quantum electrodynamics.

††journal: osajournal††articletype: Research Article

##  1 Introduction

Single photon state is a fundamental concept in quantum physics that has been
actively explored for their potential applications in quantum computing [1, 2,
3, 4, 5], quantum information processing [6, 7, 8, 9], and strongly correlated
man-body quantum phenomena [10, 11, 12]. One of the critical underlying
techniques for generating single photon emission is to achieve photon blockade
(PB), which implies that the excitation of the first photon will block the
transmission of the later arriving photons, resulting in an orderly output of
photon stream with sub-Poissonian statistics [13, 14]. Up to now, the typical
mechanism for realizing PB includes conventional PB [15] and unconventional PB
[16, 17], which rely on the strong coupling induced strong energy-spectrum
anharmonicity in the emitter-cavity system and the quantum interference
between the different excitation pathways, respectively. The above two
mechanisms for generation of strong PB have been investigated theoretically
and experimentally on various platforms based on the Jaynes-Cummings model of
a two-level system, e.g., quantum dot- and atom-cavity systems [18, 19, 20,
21, 22, 17], optomechanical systems [23, 24, 25, 26], and waveguide- and
circuit-QED systems [27, 28, 29, 30, 31, 32, 33, 34]. In particular,
unconventional PB has been experimentally demonstrated in single quantum dot-
cavity system and two coupled superconducting resonators [35, 36] by utilizing
the constructive quantum interference to suppress the two-photon excitation.
However, our recent work shows that the quantum interference mechanism for
generating strong PB [$g^{(2)}(0)\sim 0$] is still a challenge for the weak
atom-cavity coupling regime [37], which indicates that sufficient energy-
spectrum anharmonicity is necessary for most of the systems. In general,
multiphoton excitation plays an important role in the system with weak energy-
spectrum anharmonicity, although the two-photon excitation could be completely
suppressed by quantum interference for unconventional PB.

On the other hand, electromagnetically induced transparency (EIT) with its
intriguing loss free dark-state polariton (DSP) [38] has been observed in
atom-cavity systems [39, 40, 41, 42, 43], which offers even further
opportunities for realizing a large and tunable nonlinearity from the control
of light with light, i.e., giant Kerr effect [13, 14]. Moreover, remarkable
progress on realization of single atom EIT in a high-finesse optical cavity
[44, 45, 46] could allow for novel applications ranging from quantum computing
[47, 48, 49, 50, 51, 52] to quantum simulation [53] and coherent quantum
networks [54, 55]. Thus, it is natural to wonder whether strong PB can be
realized in single atom cavity-EIT by employing the DSP, which is insensitive
to the loss of the atomic excited state and possesses a significantly enhanced
coherent lifetime of quantum state [56]. An affirmative answer in exploring
this question will significantly enhance our exciting insights in quantum
physics, as well as potential novel applications in quantum information
science, where an ideal single photon source is achieved by combining the
advantages of strong nonlinearity and DSP in cavity-EIT.

In this work, we investigate the realization of strong PB by utilizing an
optical Stark shift in single alkaline-earth-metal atom EIT placed inside a
high-finesse cavity. It is shown that the combination of EIT hosted strong
nonlinearity and Stark shift enhanced energy-spectrum anharmonicity gives rise
to the strong PB, which, in the mean time, maintains a high transmission
ascribed to the atomic quasi-dark state resonance in EIT. Compared with the
conventional proposal of generating strong interacting photons in cavity
coupled condensates [14], the PB in our single atom-cavity-EIT system depends
on the optimal parameters of the Stark shift and Rabi frequency of the
classical control field. Our results reveal that a large enough Stark shift
and control field are not always beneficial for strong PB, which is different
from the well-known mechanism for generating PB with giant Kerr nonlinearity
in cavity-EIT. In particular, the optimal second-order correlation function
[$g^{(2)}(0)\sim 10^{-4}$] in our model can be reduced more than $45$ times in
magnitude compared with the extensively studied single four-level atom-cavity-
EIT with the effective Kerr nonlinearity at a moderate Stark shift [57, 58,
59, 60].

##  2 Model and experimental feasibility

Figure 1: (color online). (a) Schematic diagram of the relevant transitions in
cavity-EIT system. A single 87Sr four-level atom is trapped inside a high-
finesse optical cavity. The bias field ${\bf B}$ parallels to the cavity axis,
which defines the quantization axis and generates a Zeeman splitting
$\hbar\Delta$ between the magnetic sublevels $|3\rangle$ and $|4\rangle$ of
${}^{3}P_{1}$. The cavity is driven by a weak $\sigma$-polarized laser field
(given by the superposition of $\sigma^{+}$ and $\sigma^{-}$ polarization) and
the atom is pumped by a $\pi$-polarized classical control field orthogonal to
the cavity axis. (b) The typical energy spectrum for optical Stark shift
mediated cavity-EIT. The dressed-state splitting $\delta_{1,-}$ (the red
dashed line), $\delta_{1,0}$ (the black line) and $\delta_{1,+}$ (the blue
dotted line) as a function of the optical Stark shift $U_{0}$ for
$\Omega/g=0.01$ (c) and $\Omega/g=2.1$ (d), respectively.

We consider a single alkaline-earth-metal (like) atom inside a high-finesse
cavity subjected to a bias magnetic field ${\bf B}$ along the cavity axis
(quantization axis). Figure. 1(a) shows the level structure of the electronic
${}^{1}S_{0}$ ($F=9/2$) state to ${}^{3}P_{1}$ ($F^{\prime}=11/2$) state
transition for the 87Sr atom, in which the four relevant magnetic Zeeman
sublevels with two ground states $|F=9/2,m_{F}=9/2\rangle$ ($|1\rangle$) and
$|F=9/2,m_{F}=7/2\rangle$ ($|2\rangle$), two long-lived electronic orbital
states $|F^{\prime}=11/2,m^{\prime}_{F}=7/2\rangle$ ($|3\rangle$) and
$|F^{\prime}=11/2,m^{\prime}_{F}=11/2\rangle$ ($|4\rangle$) are included. In
our configuration, the cavity field only supports a $\sigma$-polarized
(orthogonal to the bias magnetic field ${\bf B}$) photon, which can be
decomposed into $\sigma^{-}$\- and $\sigma^{+}$-polarized modes [61]. The
dipole-forbidden singlet ${}^{1}S_{0}$ to triplet ${}^{3}P_{1}$ optical
transition is coupled to a $\sigma$-polarized cavity field with a natural
narrow linewidth of $7.5$ kHz [62]. During this coherent process, the magnetic
quantum numbers of the electronic states induced by the $\sigma^{+}$\- and
$\sigma^{-}$-polarized cavity field satisfy $m^{\prime}_{F}-m_{F}=\pm 1$.
Specifically, the single atom is initially prepared in the state $|1\rangle$.
The transition $|1\rangle\leftrightarrow|3\rangle$ at a wavelength $698$ nm is
resonantly driven by the $\sigma^{-}$-polarized optical cavity, corresponding
to the single atom-cavity coupling $g$ and the intrinsic cavity decay rate
$\kappa$. Besides the resonant transition $|1\rangle\leftrightarrow|3\rangle$,
the $\sigma^{+}$-polarized cavity field also drives the far-resonance atomic
transition $|1\rangle\leftrightarrow|4\rangle$ with the single atom-cavity
coupling $g_{0}$ and the atom-cavity detuning $\Delta$. Here the tunable
$\hbar\Delta$ denotes the Zeeman splitting between
$|F^{\prime}=11/2,m^{\prime}_{F}=7/2\rangle$ and
$|F^{\prime}=11/2,m^{\prime}_{F}=11/2\rangle$ of the ${}^{3}P_{1}$ state. A
tunable classical $\pi$-polarized (along the ${\bf B}$ field) control field
propagating orthogonal to the cavity axis drives the resonantly atomic
transition $|2\rangle\leftrightarrow|3\rangle$ with Rabi frequency $\Omega$.
In addition, the cavity is driven by a weak $\sigma$-polarized (orthogonal to
the ${\bf B}$ field) laser field with frequency $\omega_{p}$ and the driven
amplitude $\eta$ that related to the input laser power $P$ and optical-cavity
decay rate $\kappa$ by $\eta=\sqrt{2P\kappa/\hbar\omega_{p}}$, which yields
the cavity-light detuning $\Delta_{c}=\omega_{c}-\omega_{p}$ with
$\omega_{c}$, being the bare cavity frequency.

In the limit of large atom-cavity detuning of $|g_{0}/\Delta|\ll 1$, the
atomic excited state $|4\rangle$ can be adiabatically eliminated, which yields
a $\Lambda$-level scheme for generating cavity-EIT with the tunable optical
Stark shift $U_{0}=-g_{0}^{2}/\Delta$. By using the rotating wave
approximation, the relevant Hamiltonian of the single atom-cavity-EIT system
takes the form as

$\displaystyle{\hat{\cal H}}/\hbar=$ $\displaystyle
g(\hat{a}^{\dagger}\hat{\sigma}_{13}+\hat{a}\hat{\sigma}_{31})+\Omega(\hat{\sigma}_{23}+\hat{\sigma}_{32})+U_{0}\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{11}$
$\displaystyle+\Delta_{c}\hat{a}^{\dagger}\hat{a}+\Delta_{c}\hat{\sigma}_{33}+\Delta_{c}\hat{\sigma}_{22}+\eta(\hat{a}^{\dagger}+\hat{a}),$
(1)

where $\hat{a}^{\dagger}$ ($\hat{a}$) is the creation (annihilation) operator
of the single mode cavity, $\hat{\sigma}_{ij}$ are the atomic spin projection
operators with $i,j=1,2,3$ labeling the internal states of the atom. For
simplicity, the two-photon detuning of the Raman process is fixed equal to the
cavity-light detuning $\Delta_{c}$ by tuning the control field.

Compared with the experimental studies on single atom-cavity-EIT in Ref. [44],
the additional optical Stark shift in Eq. (2) plays an important role in
changing the energy spectrum of the cavity-EIT system, especially controlling
the quantum statistical properties of the transmission photons. In addition,
the term of $U_{0}\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{11}$ in our model
could facilitate more for the emergence of strong PB at a moderate strength of
$U_{0}$ in contrast to the proposals in single four-level atom-cavity-EIT with
the Stark shift term of $U_{0}\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{22}$ [57,
58, 59, 60], as we shall see below.

To generate the large $U_{0}$, the condition of $|g_{0}/g|^{2}\gg 1$ should be
satisfied. Therefore, the matrix element of atomic transition
$|1\rangle\leftrightarrow|3\rangle$ should be much smaller than
$|1\rangle\leftrightarrow|4\rangle$. In our configuration in Fig. 1(a), we
find that the atom-cavity coupling strength ratio between the $\sigma^{+}$ and
$\sigma^{-}$ polarization of the cavity field is $|g_{0}/g|^{2}=165/3\gg 1$ by
calculating the relevant transition matrix elements. In fact, the analogous
proposal by utilizing dipole-forbidden transition has been realized
experimentally in cavity enhanced magnetically induced optical transparency on
88Sr atomic ensembles in Ref. [63]. Moreover, we should note that the
elaborately selective long-lived state in our model can also be employed by
utilizing the clock transition, owing to the unique energy-level structures
for alkaline-earth-metal atoms [64, 65, 66].

##  3 Energy spectrum

To investigate the underlying physical mechanism for photon quantum statistics
with combining the cavity-EIT induced nonlinearity and Stark shift, we
calculate the energy spectrum of the system by diagonalizing the Hamiltonian
in Eq. (2) with ignoring the weak driving field of the cavity. The total
excitation numbers for the single $\Lambda$-level atom trapped in the cavity-
EIT system is conserved with $\eta=0$. Then, the accessible Hilbert space can
be restricted to Fock states with the type $|n,N_{1},N_{2},N_{3}\rangle$,
where $N_{i}(i=1,2,3)$ represents the number of atoms trapped in $|i\rangle$
state and $n$ denotes the photon number for the cavity. Explicitly, the
relevant three spanned states are $|n,1,0,0,\rangle$, $|n-1,0,1,0\rangle$, and
$|n-1,0,0,1\rangle$. Therefore, the corresponding block can be written in
matrix form in the basis ${\hat{\cal H}}\Psi={\cal M}\Psi$ with
$\Psi=[|1,0,0,n\rangle,|0,1,0,n-1\rangle,|0,0,1,n-1\rangle]^{T}$. The matrix
${\mathcal{M}}$ is obtain as

$\displaystyle{\mathcal{M}}=\left(\begin{array}[]{ccc}n\Delta_{c}+nU_{0}&0&g\sqrt{n}\\\
0&n\Delta_{c}&\Omega\\\ g\sqrt{n}&\Omega&n\Delta_{c}\\\ \end{array}\right)$
(5)

by diagonalizing the matrix in Eq. (5), the energy spectrum for the nonzero
photon excitation recasts into three branches. Fig. 1(b) shows the typical
anharmonicity ladder of the energy spectrum. The asymmetric energy splitting
of the $n$th dressed states induced by the Stark shift for the higher- (
$|n,+\rangle$ ), middle- ( $|n,0\rangle$ ) and lower- ( $|n,-\rangle$ )
branches are labeled as $\delta_{n,+}$, $\delta_{n,0}$, and $\delta_{n,-}$,
respectively. For $U_{0}=0$, the middle-branch splitting $\delta_{n,0}=0$,
which exhibits a DSP that denotes atomic dark-state resonance. It is obvious
that the DSP vector can be written as

$\displaystyle|n,0\rangle=\beta_{+}|n\rangle|1\rangle+\beta_{0}|n-1\rangle|3\rangle+\beta_{-}|n-1\rangle|2\rangle,$
(6)

with $\beta_{+}=-\Omega/\sqrt{\Omega^{2}+g^{2}n}$,
$\beta_{-}=g\sqrt{n}/\sqrt{\Omega^{2}+g^{2}n}$, and $\beta_{0}=0$. However,
this dark-state resonance should be strictly called “quasi-dark-state
resonance” in the presence of nonzero $U_{0}$, where the DSP is broken with
$\beta_{0}\neq 0$.

Figures. 1(c) and 1(d) show the dressed-state splitting $\delta_{1,\nu}$ as a
function of Stark shift $U_{0}$ for weak ($\Omega/g=0.01$) and strong
($\Omega/g=2.1$) control fields by fixing $g/\kappa=4$, respectively, where
$\nu=\pm 1,0$ represents the upper, lower, and middle helicity branches,
respectively. An asymmetry structure of the dressed-state splitting between
the red and blue light-cavity detuning is observed, where the result is
analogous to our recent study on the Stark shift mediated Jaynes-Cummings
model [37]. Moreover, the $U_{0}$-dependent lower branch splitting
$\delta_{1,-}$ is significantly enhanced with the increasing negative
$|U_{0}|$, while gradually saturated to a constant for positive $U_{0}$, vice
verse for the case of the higher branch splitting $\delta_{1,+}$.

Interestingly, the middle-branch splitting at atomic quasi-dark-state
resonance is insensitive to $U_{0}$ with roughly zero shift ($\delta_{1,0}\sim
0$) in the presence of a weak control field. But, for a large control field,
the value of $\delta_{1,0}$ enhances rapidly with the increasing of both
positive and negative Stark shift $|U_{0}|$. As expected, the observed higher
anharmonic ladder of the energy spectrum could facilitate the generation of
strong nonclassical photons by employing the enhanced nonlinearity in Stark
shift mediated cavity-EIT. In the following, we will show the strong PB
induced by combining optical Stark shift $U_{0}$ and the loss-insensitive
atomic quasi-dark-state resonance in the EIT-cavity system.

##  4 Numerical results

The quantum statistical properties of photons in the cavity are characterized
by numerically solving the quantum master equation using Quantum Optics
Toolbox [67], which takes into account the dissipation of photons and atom.
The atom-cavity system can be described by the density matrix $\rho$ for the
cavity mode and the atomic field. Then the time evolution of the density
matrix $\rho$ obeys the master equation, $\frac{d\rho}{dt}={\cal{L}}\rho$ with
the Liouvillian superoperator defined as

${\cal{L}}\rho=-i[\hat{H},{\rho}]+\frac{\kappa}{2}\mathcal{\cal{D}}[\hat{a}]\rho+\frac{\gamma}{2}\mathcal{\cal{D}}[\hat{\sigma}_{13}]\rho+\frac{\gamma}{2}\mathcal{\cal{D}}[\hat{\sigma}_{23}]\rho,$
(7)

where
$\mathcal{D}[\hat{o}]\rho=2\hat{o}{\rho}\hat{o}^{\dagger}-\hat{o}^{\dagger}\hat{o}{\rho}-{\rho}\hat{o}^{\dagger}\hat{o}$
denotes the Lindblad type of dissipation. Thus, the steady-state intracavity
photon number ${n}_{s}={\rm Tr}(\hat{a}^{\dagger}\hat{a}\rho_{s})$ is
evaluated by solving the steady-state master equation with ${\cal
L}\rho_{s}=0$. It is straightforward to obtain the cavity transmission
$T_{a}=n_{s}/n_{0}$ with $n_{0}=(\eta/\kappa)^{2}$ being the bare cavity
photon number. In addition, the general time interval $\tau$ dependence
second-order correlation function for isolated photons is defined as

$\displaystyle g^{(2)}(\tau)=\frac{{\rm
Tr}[\hat{a}^{\dagger}(t)\hat{a}^{\dagger}(t+\tau)\hat{a}(t+\tau)\hat{a}(t)\rho_{s}]}{{\rm
Tr}[\hat{a}^{\dagger}(t)\hat{a}(t)\rho_{s}]^{2}}.$ (8)

For $\tau=0$, $g^{(2)}(\tau)$ is directly reduced to the equal time second-
order correlation function $g^{(2)}(0)$. As a result, the photon quantum
statistics for PB should satisfy both the subPoissonian statistics with
$g^{(2)}(0)<1$ and photon antibunching with $g^{(2)}(0)<g^{(2)}(\tau)$,
simultaneously. Without loss of generality, the strong PB is defined as
$g^{(2)}(0)<0.01$, where photon antibunching is confirmed by using the quantum
regression theorem [37].

In our numerical simulation, we take the cavity decay rate $\kappa=2\pi\times
160$ kHz, which has been realized in recent experiment of superradiance for
87Sr clock transition [62], the atomic spontaneous decay rate
$\gamma=2\pi\times 7.5$ kHz for the long-lived excited state of ${}^{3}P_{1}$,
the weak cavity driven strength $\eta/\kappa=0.1$, and the single atom-cavity
coupling $g/\kappa=4$, for which coupling strength on a forbidden transition
has demonstrated capability in current experiments for single atoms trapped in
the cavity with the quantization length of the cavity around a few millimeters
[63, 15]. Therefore, the free parameters in our system are reduced to cavity-
light detuning $\Delta_{c}$, Rabi frequency of the control field $\Omega$, and
optical Stark shift $U_{0}$ by continuously tuning the sign and strength of
Zeeman splitting $\hbar\Delta$ in experiment [63]. Limited by the validity of
our model, the numerical results presented below exhibit the parameter space
$0\leq U_{0}/g\leq 4$, corresponding the tunable magnetic field around a few
tens of Gauss. Finally, we point out that our scheme should also be applicable
to other alkaline-earth-metal atoms, e.g., Yb atom [68], which also possess
the long-lived excited states.

Figure 2: (a) Second-order correlation function $g^{(2)}(0)$ and (b) cavity
transmission $T_{a}$ as a function of cavity-light detuning $\Delta_{c}$ for
$U_{0}/g$= 0 and 1. (c) $g^{(2)}(0)$ and (d) $T_{a}$ as a function of
$\Delta_{c}$ and optical Stark shift $U_{0}$. The white dashed line shows the
corresponding dressed state splitting for the three branches of the energy
spectrum. The colors with blue-red gradient shading indicate the values of
log$[g^{(2)}(0)]$ in (c) and $T_{a}$ in (d).

In the weak control field limit, e.g., $\Omega/g=0.01$, we plot the second-
order correlation function at zero time delay $g^{(2)}(0)$ and the cavity
transmission $T_{a}$ as a function of cavity-light detuning $\Delta_{c}$ for
different optical Stark shift $U_{0}$, as shown in Figs. 2(a) and 2(b). As can
be seen, a symmetry structure for $g^{(2)}(0)$ with two minimum values
appearing at the red and blue sidebands with $\Delta_{c}=\delta_{1,\pm}$ is
observed in absence of the Stark shift. Besides, a narrow EIT transmission
window appears at atomic dark-state resonance with $\Delta_{c}=\delta_{1,0}$,
which emerges due to the quantum interference between the cavity field and the
control light. In particular, the nearly $100\%$ cavity transmission maintains
a coherent quantum statistic by keeping $g^{(2)}(0)=1$ when $U_{0}=0$. As
expected, the symmetry structures for $g^{(2)}(0)$ and $T_{a}$ between the red
and blue sidebands are broken when the optical Stark shift $U_{0}$ are
introduced. However, the cavity transmission $T_{a}$ and photon quantum
statistics $g^{(2)}(0)$ are insensitive to $U_{0}$, although the Stark shift
obviously changes the vacuum-Rabi splitting at the red and blue sidebands of
the system.

Figures. 2(c) and 2(d) illustrate the results for log[$g^{(2)}(0)$] and
$T_{a}$ as a function of $U_{0}$ and $\Delta_{c}$. It is shown that
$g^{(2)}(0)$ and $T_{a}$ both depend on the signs of $U_{0}$ and $\Delta_{c}$,
and the smaller value of $T_{a}$ corresponds to the smaller $g^{(2)}(0)$ at
the red (blue) sideband of vacuum-Rabi resonance for positive (negative)
$U_{0}$. The white dashed lines show the corresponding dressed-state splitting
of the three branches by analytically calculating the energy spectrum, which
are highly consistent with the numerical results by solving the master
equation in Eq.(7) as well. In contrast to the spin-$1/2$ model [37], the
$g^{(2)}(0)$ does not exhibit a significant reduction with the enhanced PB,
even for a large $|U_{0}|$. It is understood that the quantum interference
mechanism for realizing strong PB is absent in our cavity-EIT model at the
vacuum-Rabi splitting ($\Delta_{c}=\delta_{1,\pm}$), which reveals that the
Stark shift enhanced energy-spectrum anharmonicity is not sufficient to
guarantee the strong PB.

Figure 3:  (a) $g^{(2)}(0)$ and (b) $T_{a}$ as a function of $\Delta_{c}$ with
$\Omega/g=2$ for $U_{0}/g$=0 and 1.6. The solid blocks mark the minimum value
of $g^{(2)}(0)$ for $U_{0}/g$=1.6 and the corresponding $T_{a}$.
Log$[g^{(2)}(0)]$ (c) and $T_{a}$ (d) as a function of $\Delta_{c}$ and
$\Omega$. The white dashed line shows the corresponding dressed state
splitting for the three branches of the energy spectrum. The optimal
$g^{(2)}_{\rm{opt}}(0)$ (e) and the corresponding cavity transmission
$T_{\rm{opt}}$ (f) as a function of $\Omega$. In (c)-(d), the other parameter
is fixed at $U_{0}/g=1.6$.

To proceed further, we study the photon quantum statistics $g^{(2)}(0)$ and
cavity transmission $T_{a}$ in a large control field by fixing $\Omega/g=2$.
As shown in Figs. 3(a) and 3(b), compared with the weak control field case, we
find that $T_{a}$ decreases rapidly with the increasing $\Omega$ at the red
and blue sidebands of vacuum-Rabi resonance for $U_{0}/g=0$ (blue dashed
line), although the corresponding $g^{(2)}(0)$ has a slight decrease due to
the large control field enhanced cavity-EIT nonlinearity. But, for nonzero
Stark shift with $U_{0}/g=1.6$ (red solid line), we notice that the photon
quantum statistics host a strong PB with $g^{(2)}(0)=3.6\times 10^{-3}$ at
atomic quasi-dark resonance with $\delta_{1,0}=-1.16g$ in Fig. 3(a). In
particular, the output of the cavity filed still exhibits a high transmission
$T_{a}$, which is apparently larger than the value of $T_{a}$ at the red
sideband of vacuum-Rabi splitting [Fig. 3(b)]. Therefore, the realization of
strong PB by utilizing atomic quasi-dark resonant with a high transmission is
of great benefit to generate an ideal single photon source in cavity-EIT. We
should note that the predicted narrow transmission transparency window
employing advantages of cavity-EIT may facilitate the studies of high quality
single atom transistors and all-optical switching in single photon levels [69,
47, 70, 48].

Figures. 3(c) and 3(d) display the contour plots of $g^{(2)}(0)$ and $T_{a}$
on the $\Delta_{c}$-$\Omega$ parameter plane for $U_{0}/g=1.6$. As can be
seen, atomic quasi-dark resonance with strong PB is observed by tuning the
Rabi frequency $\Omega$ of the control field. The dressed state splitting for
the three branches exhibits the large nonlinear shifts with the increasing
$\Omega$, which also agrees with the analytically asymmetric energy splitting
for the first manifold in energy spectrum (white dashed lines). In order to
quantitatively characterize the quantum statistics for the cavity field, we
plot the minimum $g^{(2)}_{\rm{opt}}(0)=\min[g^{(2)}(\Delta_{c})]$ and the
corresponding cavity transmission $T_{\rm{opt}}$ at the same value of the
cavity-light detuning versus $\Omega$, as displayed in figs. 3(e) and 3(f). It
is clear that both $g^{(2)}_{\rm{opt}}(0)$ and $T_{\rm{opt}}$ possess a dip
feature, corresponding the local minimum with $g^{(2)}(0)=3.5\times 10^{-3}$
and $T=0.55$ for different $\Omega$. Interestingly, the optimal PB with the
optical control field $\Omega/g=2.1$ corresponds to a large cavity transition
up to $T=0.87$, which is crucial for applications of high-quality single
photon sources.

Figure 4: Contour plots of log$[g^{(2)}(0)]$ (a) and cavity transmission (b)
as a function of $\Delta_{c}$ and $U_{0}$. The white dashed line shows the
dressed state splitting for the three branches of energy spectrum. The optimal
$g^{(2)}_{\rm{opt}}(0)$ (c) and the corresponding $T_{\rm{opt}}$ [the red
curve] (d) as a function of $U_{0}$, and the light pink area exhibit the
strong PB regime and the corresponding $T_{a}$, respectively. The black curve
in (d) shows the corresponding occupy probability of atomic excited state
$|3\rangle$. Here, the classical control field is fixing at the optimal value
with $\Omega/g=2.1$.

To gain more insight into the PB in the cavity-EIT, we further calculate the
$g^{(2)}(0)$ and $T_{a}$ in the $\Delta_{c}$-$U_{0}$ parameter plane by fixing
the optimal control field $\Omega/g=2.1$, as shown in Figs. 4(a) and 4(b).
Compared with the results of the weak control field shown in Figs. 2(c) and
2(d), the photon quantum statistics and cavity transition are highly sensitive
to the strength and sign of the optical Stark shift for the large control
field. In particular, the strong PB occurs at the middle branch accompanied
with atomic quasi-dark resonance. With the increasing of $U_{0}$, the optimal
$g^{(2)}_{\rm{opt}}(0)$ displays a dip structure, which is dramatically
decreasing from one to a local minimum with a optimal
$g^{(2)}_{\rm{opt}}(0)=9.0\times 10^{-5}$ at $U_{0}/g=3.0$ [Fig. 4(c)]. The
typical feature of $T_{\rm{opt}}$ is gradually decreasing as a function of
$U_{0}$, corresponding the probability of the excited state $|\beta|^{2}$ in
Eq. (6) that is gradually growing at the atomic quasi-dark resonance [Fig.
4(d)]. We also checked that the photon antibunching $g^{(2)}(0)$ gradually
increases with the driven amplitude $\eta$ of the input laser, albeit the
steady-state intracavity photon number $n_{s}$ is obviously growing.

Remarkably, the optimal $g^{(2)}(0)$ for strong PB emerging at atomic quasi-
dark resonance is more than four orders of magnitude smaller than the case of
coherent EIT transition in the absence of $U_{0}$. We emphasize that the
strong PB is contributed by combination of the Stark shift enhanced energy-
spectrum anharmonicity and EIT hosted large nonlinearity at atomic quasi-dark
state resonance. In addition, the strong PB with a large cavity transmission
exists in a large parameter region, even deviating from the optimal parameters
[the light pink area in Fig. 4(c)], which could facilitate the experimental
feasibility for studying the exotic nonclassical quantum states in cavity-EIT.

Compared with the Stark shift term of
$U_{0}\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{11}$ in Eq. (2), the reduced Stark
shift for the typical scheme of a single four-level atom-cavity-EIT [57, 58,
59, 60] takes the form of $U_{0}\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{22}$, in
which the cavity field drives the far-resonance atomic
$|2\rangle\leftrightarrow|4\rangle$ transition. Therefore, the realization of
PB in these proposals [57, 58, 60, 59, 71, 72] can be ascribed to the large
nonlinearity from generating the Kerr effect. However, the strong PB in our
model with existing the optimal optical Stark shift and the classical control
field announces that the mechanism for generating strong PB cannot be
completely ascribe to the Kerr nonlinearity in cavity-EIT [13, 14].

Figure 5: (a) Contour plots of log$[g^{(2)}(0)]$ for the earlier proposals in
Ref. [57, 58, 59, 60] on the $U_{0}$-$\Omega$ parameter plane with
$\Delta_{c}/g=0$. The light pink area exhibiting the strong PB regime with
($g^{(2)}(0)<0.01$) is a guide for the eye. (b) The optimal
$g^{(2)}_{\rm{opt}}(0)$ in the presence of Stark shift
$U_{0}\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{11}$ (dashed line) with
$\Omega/g=2.1$ and $U_{0}\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{22}$ (solid
line) with $\Omega/g=0.44$ as a function of $U_{0}$, respectively.

To further quantitatively distinguish the essential differences of the two
proposals, we calculate $g^{(2)}(0)$ in the presence of Stark shift term of
$U_{0}\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{22}$ in the $U_{0}$-$\Omega$
parameter plane, as displayed in Fig. 5(a). Here, the cavity-light detuning is
fixed at $\Delta/g=0$, since the optimal PB occurred at the position of atomic
dark-state resonance for atom-cavity EIT. For a fixed $|U_{0}|$, the minimum
photon antibunching amplitude of $g^{(2)}(0)$ corresponds to the optimal
control field with $\Omega/g=0.44$. It is clear that the optimal
$g^{(2)}_{\rm{opt}}(0)$ is monotonically decreased with the increasing of the
effective Kerr nonlinearity by tuning $U_{0}$. In order to compare the PB
between our proposal and earlier schemes [57, 58, 59, 60] more intuitively,
the optimal $g^{(2)}_{\rm{opt}}(0)$ as a function of $U_{0}$ is illustrated,
as shown in Fig. 5(b). As can be seen, the optimal value of $g^{(2)}(0)$ in
our scheme is much less than the previous proposals from hosting a wide range
parameter regime of strong PB. In particular, $g^{(2)}(0)$ in our scheme can
be reduced more than $45$ times in magnitude at a moderate Stark shift with
$U_{0}/g=3.0$, where the strong antibunching photons are ascribed to the
significantly enhanced energy-spectrum anharmonicity [Fig. 1(c)] induced by
$U_{0}$.

Furthermore, the essential difference between the two schemes for generating
PB can be readily understood by comparing the energy spectrum of the system.
For the single four-level atom-cavity-EIT in Refs. [57, 58, 59, 60], we find
that the first dressed state in the energy spectrum is immune to the changing
of $U_{0}$. Therefore, the vacuum-Rabi splitting
$\delta_{1,\pm}=\pm\sqrt{g^{2}+\Omega^{2}}$ and $\delta_{1,0}=0$ for the three
branches are all independent of $U_{0}$. From this perspective, the Stark
shift enhanced vacuum-Rabi splitting in our proposal is equivalent to
enlarging the effective single atom-cavity coupling. From the experimental
point of view, the realization of strong PB beyond the strong optical Stark
shift in our scheme could obviously improve the accessibility and
controllability in the experiment. Therefore, our proposal can be used to
realize a high-quality single photon source for a single atom-cavity-EIT
mediated by a moderate optical Stark shift. We should note that the advantage
of generating PB based on Kerr nonlinearity is using an ultracold ensemble
coupled to the cavity [13, 14], where the giant Kerr nonlinearity could be
emerged on the atomic dark state resonance of cavity-EIT.

##  5 Conclusions

In summary, we propose an experimental scheme for generating PB in optical
Stark shift mediated-cavity-EIT by employing the advantages of energy-level
structures in single alkaline-earth-metal atoms. By utilizing atomic quasi-
dark state resonance, strong PB with second-order correlation function and
high cavity transmission are achieved by tuning the optimal Stark shift and
control field. The mechanism in our model is owe to the combination of EIT
hosted large nonlinearity and Stark shift enhanced energy-spectrum
anharmonicity, which provides new insights to the PB mechanism in cavity-EIT
beyond the giant Kerr nonlinearity. Importantly, all parameters for our
numerical simulation are based on the current experimental capabilities.
Compared with the traditional single four-level atom-cavity-EIT systems [57,
58, 59, 60], the strong PB with $g^{(2)}(0)=9\times 10^{-5}$ can be achieved
at a moderate Stark shift, which could significantly facilitate the
realization of strong PB beyond the strong atom-cavity coupling.

Meanwhile, our scheme has the following advantages in contrast to the Stark
shift mediated Jaynes-Cummings model [37]. (i) Our scheme does not require the
quantum interference conditions, which rely on the precise control of the
phase and the coupling strength of an external microwave field simultaneously.
Indeed, the quantum interference conditions are difficult to implement for a
complex multilevel atom-cavity system. (ii) By utilizing the atomic quasi-
dark-state resonance of cavity-EIT, the large cavity transmission can be
expected, which is insensitive to the inevitable spontaneous emission of
excited states. (iii) In our configuration, the photon antibunching amplitude
of $g^{(2)}(0)$ can be reduced more than one order of magnitude compared with
the proposal of quantum interference at a moderate Stark shift. In fact, we
should emphasize that the multiphoton ($n\geq 3$) excitations are hard to be
suppressed for the quantum interference mechanism, although the two-photon
excitation is completely eliminated. Finally, our experimental proposal for
generation of high-quality single photon sources could implicate exciting
opportunities for potential applications in quantum information science and
quantum metrology [73].

## Fundings

This work is supported by the National Natural Science Foundation of China
(11804409, 11874433, 12025509, 11874434), the National Key R$\&$D Program of
China (2018YFA0307500), the Fundamental Research Funds for the Central
Universities (Grant No. 18lgpy80), and the Key-Area Research and Development
Program of Guang-Dong Province under Grants No. 2019B030330001.

## Disclosures

The authors declare no conflicts of interest.

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