# Strong Photon Blockade Mediated by Optical Stark Shift in a Single-Atom-
Cavity System

Jing Tang  Laboratory of Quantum Engineering and Quantum Metrology, School of
Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082,
People’s Republic of China  Yuangang Deng<EMAIL_ADDRESS>Laboratory
of Quantum Engineering and Quantum Metrology, School of Physics and Astronomy,
Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, People’s Republic of
China  Chaohong Lee<EMAIL_ADDRESS>Laboratory of Quantum
Engineering and Quantum Metrology, School of Physics and Astronomy, Sun Yat-
Sen University (Zhuhai Campus), Zhuhai 519082, People’s Republic of China
State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen
University (Guangzhou Campus), Guangzhou 510275, People’s Republic of China
Synergetic Innovation Center for Quantum Effects and Applications, Hunan
Normal University, Changsha 410081, People’s Republic of China

###### Abstract

We propose a theoretical scheme to achieve strong photon blockade via a single
atom in cavity. By utilizing optical Stark shift, the dressed-state splitting
between higher and lower branches is enhanced, which results in significant
increasing for lower (higher) branch and decreasing for higher (lower) branch
at the negative (positive) Stark shift, and dominates the time-evolution of
photon number oscillations as well. Furthermore, the two-photon excitation is
suppressed via quantum interference under optimal phase and Rabi frequency of
an external microwave field. It is shown that the interplay between the
quantum interference and the enhanced vacuum-Rabi splitting gives rise to a
strong photon blockade for realizing high-quality single photon source beyond
strong atom-cavity coupling regime. In particular, by tuning the optical Stark
shift, the second-order correlation function for our scheme has three orders
of magnitude smaller than Jaynes-Cummings model and correspondingly there
appears a large cavity photon number as well. Our proposal may implicate
exciting opportunities for potential applications on quantum network and
quantum information processing.

##  I INTRODUCTION

The realization and manipulation of nonclassical light fields at single-photon
level plays an essential role in quantum information science Duan  _et al._
(2001); Duan and Kimble (2004); Kok  _et al._ (2007); Kimble (2008);
Giovannetti  _et al._ (2011). It offers unprecedented scientific opportunities
for exploring various applications in quantum computing Bennett and DiVincenzo
(2000); Knill  _et al._ (2001); Buluta  _et al._ (2011), quantum key
distribution Scarani  _et al._ (2009); O’Brien  _et al._ (2009), quantum
simulation Buluta and Nori (2009); Georgescu  _et al._ (2014), and quantum
metrology Giovannetti  _et al._ (2011). Recently, the controllable single-
photon emitters have been extensively explored in different quantum systems,
including atom-cavity coupled systems McKeever  _et al._ (2004); Wilk  _et
al._ (2007); Le Boité  _et al._ (2016), quantum-dot-cavity coupled systems
Hennessy  _et al._ (2007); Toishi  _et al._ (2009); Müller  _et al._ (2015a),
coupled optomechanical systems Rabl (2011); Nunnenkamp  _et al._ (2011); Wang
_et al._ (2015), coupled optical-fiber systems Xu  _et al._ (2007, 2008),
waveguide-QED systemsMirza and Schotland (2016a); Mirza  _et al._ (2017);
Gheeraert  _et al._ (2018); Zheng  _et al._ (2011); Mirza and Schotland
(2016b), and circuit-QED systems Hoffman  _et al._ (2011); Lang  _et al._
(2011); Liu  _et al._ (2014), in which the quantum statistics of the output
photons show a sub-Poissonian distribution due to the photon blockade (PB) Zou
and Mandel (1990).

There are two typical PB mechanisms that can be used for generating a single
photon in cavity QED. One is based on strong coupling with a large vacuum Rabi
splitting in a cavity with a single two-level atom Imamoǧlu  _et al._ (1997);
Hartmann and Plenio (2007); Chang  _et al._ (2014), which gives rise to an
anharmonic ladder of dressed states in the energy spectrum. When the energy
gap of the two-photon resonance is larger than the cavity decay rate, a first
photon generated in the cavity will block the transmission of a second photon
Imamoǧlu  _et al._ (1997). This mechanism relying on strong energy-spectrum
anharmonicity, is referred to as conventional PB, and remains a great
challenge, despite some experimental advances in realizing strong coupling in
a high-finesse cavity Birnbaum  _et al._ (2005); Hartmann  _et al._ (2006);
Fink  _et al._ (2008); Faraon  _et al._ (2008); Dayan  _et al._ (2008); Chang
_et al._ (2008); Brossard  _et al._ (2010); Qian  _et al._ (2018, 2019).

Figure 1: (a) Schematic illustration of creation of PB in a cavity with a
single three-level atom. In order to realize quantum interference, a microwave
field is employed to couple the spin states $|\uparrow\rangle$ and
$|\downarrow\rangle$. (b) Typical anharmonic energy spectrum with ignorable
weak driving and microwave fields. Here, $\Delta_{n,\pm}$ is the energy
splitting for the n-$th$ excited state, defined by Eq. (2). (c) Vacuum-Rabi
splitting [$\Delta_{1,-}$(red line) and $\Delta_{1,+}$ (blue line)] versus
optical Stark shift $U_{0}$ for $g/\kappa=1$. The horizontal dashed lines at
$\Delta_{1,\pm}=\pm 1$ are guides to the eye.

In contrast to conventional PB, unconventional PB has led to tremendous
advances in achieving strong antibunching of photons by using quantum
interference Liew and Savona (2010); Bamba  _et al._ (2011); Majumdar  _et
al._ (2012a); Tang  _et al._ (2015); Zhou  _et al._ (2015); Flayac and Savona
(2017); Sarma and Sarma (2018); Snijders  _et al._ (2018); Vaneph  _et al._
(2018), in which the basic principle is based on constructive interference
between different quantum transition paths from the ground state to a two-
photon excitation state. Unconventional PB has been observed experimentally in
systems ranging from a single-quantum-dot cavity Snijders  _et al._ (2018) to,
currently, two coupled superconducting resonators Vaneph  _et al._ (2018).
However, the quantum interference mechanism cannot eliminate all multiphoton
excitations simultaneously, even if the two-photon excitation state is
suppressed completely. In fact, we find that the strong PB based on quantum
interference also requires sufficient energy-spectrum anharmonicity.
Therefore, it is of great interest to answer whether there exists a new system
that utilizes advantages of both conventional and unconventional PB to realize
an ideal single-photon source beyond strong coupling regime. An affirmative
answer will significantly enhance our understanding of fundamental quantum
phenomena in cavity QED and stimulate the relevant applications in quantum
information science.

In this paper, we study such a scenario in which strong PB is generated via
Raman and microwave fields with a single atom in an optical cavity. We show
that, depending on the strength and sign of the optical Stark shift, the
vacuum Rabi splitting of the system in the red or blue sideband can be largely
enhanced, and this also determines the photon-number oscillations in the time
evolution. Remarkably, strong photon antibunching with a relatively large
cavity photon number is predicted when an enhanced vacuum Rabi splitting with
generation of a strong energy-spectrum anharmonicity is combined with quantum
interference with elimination of two-photon excitation beyond the strong-
coupling regime. The proposed scheme has the advantage that it does not
require strong atom-cavity coupling, which may facilitate the study of quantum
networks and high-quality multiphoton sources. In addition, the excitation-
energy spectrum is sensitive to the sign of the optical Stark shift, which
could provide a new method for diagnosis of atomic states by means of quantum
nondemolition measurement of the statistical properties of photons.

##  II MODEL AND ENERGY SPECTRUM

We consider a single three-level atom inside an optical cavity. The atom may
occupy one excited state $|e\rangle$ and two hyperfine ground states
$|\uparrow\rangle$ and $|\downarrow\rangle$ Mücke  _et al._ (2010);
Kampschulte  _et al._ (2010). A bias magnetic field ${\bf B}$ is applied along
the cavity axis, which produces a large enough Zeeman shift $\hbar\omega_{Z}$
between the two hyperfine ground states and defines the quantization $z$ axis.
As shown in Fig. 1(a), the magnetic quantum numbers of the electronic states
satisfy $m_{\uparrow}=m_{\downarrow}+1$ and $m_{|e\rangle}=m_{\downarrow}$.
Specifically, the single atom is pumped by a $\pi$-polarized classical laser
field with frequency $\omega_{L}$ propagating orthogonal to the cavity axis,
which drives an atomic transition $|\downarrow\rangle\leftrightarrow|e\rangle$
with Rabi frequency $\Omega$ and atom-pump detuning $\Delta$. As to the
cavity, Bragg scattering into the $\pi$-polarized cavity mode (with
polarization along the cavity axis) is forbidden in the configuration of the
laser and cavity used [Fig. 1(a)]. Therefore the cavity only supports only a
$\sigma$-polarized light field originating from the vacuum-stimulated Raman
transitionDeng  _et al._ (2014). Then the atomic transition
$|\uparrow\rangle\leftrightarrow|e\rangle$ is driven by the $\sigma$-polarized
cavity field with a single-photon coupling strength $g_{0}$, where the bare
cavity frequency and decay rate are $\omega_{c}$ and $\kappa$, respectively.
In addition, the cavity is also driven by a weak laser field with frequency
$\omega_{p}$ and amplitude $\eta$.

In the limit of large atom-pump detuning, $|g_{0}/\Delta|\ll 1$ and
$|\Omega/\Delta|\ll 1$, the excited state $|e\rangle$ can be adiabatically
eliminated. Using the rotating-wave approximation, the relevant Hamiltonian of
the single-two-level-atom-cavity system is given by

$\displaystyle\hat{H}/\hbar$
$\displaystyle=\Delta_{c}\hat{a}^{\dagger}\hat{a}+(U_{0}\hat{a}^{\dagger}\hat{a}-\Delta_{a})|\uparrow\rangle\langle\uparrow|+\eta(\hat{a}^{\dagger}+\hat{a})$
$\displaystyle+[(g\hat{a}^{\dagger}+\Omega_{m}e^{i\theta})|\uparrow\rangle\langle\downarrow|+{\rm
H.c.}],$ (1)

where $\hat{a}^{\dagger}$ ($\hat{a}$) is the creation (annihilation) operator
of the single cavity mode, $g=-\Omega g_{0}/\Delta$ is the Raman coupling
strength, $U_{0}=-g_{0}^{2}/\Delta$ is the optical Stark shift,
$\Delta_{c}=\omega_{c}-\omega_{p}$ is the cavity-light detuning, and
$\Delta_{a}=\omega_{Z}+\omega_{L}-\omega_{p}-\Omega^{2}/\Delta$ is the
effective two-photon detuning. In addition, the two ground states are also
coupled by a microwave field at the Rabi frequency $\Omega_{m}e^{i\theta}$.
Here $\theta$ is the tunable phase difference between the Raman coupling and
the microwave field. As we shall see below, the additional microwave field
plays an essential role in generating a large cavity photon number and strong
PB based on the quantum interference mechanism.

In the limit of weak driving and microwave fields, $|\eta/g|\ll 1$ and
$|\Omega_{m}/g|\ll 1$, the total excitation number is conserved. Therefore the
relevant Hilbert space of the system can be restricted to $|n,\uparrow\rangle$
and $|n-1,\downarrow\rangle$, where $n$ represents the number of photons in
the cavity. Then, the Hamiltonian in Eq. (II) can be diagonalized,
corresponding to the energy eigenvalues
$E_{n\pm}=n\Delta_{c}+nU_{0}/2\pm\sqrt{n^{2}U_{0}^{2}+4ng^{2}}/2$ when we set
$\Delta_{a}=\Delta_{c}$. Here the energy splittings of the n-$th$ pair of
dressed states $|n,\pm\rangle$ are given by

$\displaystyle\Delta_{n,\pm}=[nU_{0}\pm\sqrt{n^{2}U_{0}^{2}+4ng^{2}}]/2,$ (2)

where $+$ ($-$) denotes the higher (lower) branch. It is clear that the
splittings ($\sqrt{n^{2}U_{0}^{2}+4ng^{2}}$) between the higher and lower
branches of the dressed states are enhanced (roughly linearly increased) when
the optical Stark shift $|U_{0}|$ and the excitation number $n$ are increased.
In Fig. 1(b), we show the anharmonicity ladder of the energy spectrum.
Interestingly, an asymmetry in the dressed-state splitting between red and
blue light-cavity detunings is observed for a nonzero optical Stark shift
$U_{0}$.

Remarkably, $U_{0}$ plays an important role in modifying the vacuum Rabi
splitting $|\Delta_{1,\pm}|$, which significantly increases (decreases) the
values of $|\Delta_{1,-}|$ ($|\Delta_{1,+}|$) for the lower (higher) branch
when $U_{0}<0$ ($U_{0}>0$). The variation in the vacuum Rabi splitting
$\Delta_{1,\pm}$ as a function of $U_{0}$ with $g/\kappa=1$ is displayed in
Fig. 1(c). As can be seen, $U_{0}$ gives rise to an asymmetric vacuum Rabi
splitting between red and blue light-cavity detunings. For red detuning of the
excitation, $\Delta_{1,-}$ gradually approaches to zero with increasing
positive $U_{0}$. But for negative $U_{0}$, $|\Delta_{1,-}|$ is clearly
dependent on the value of $U_{0}$, with approximately linear growth as
$|U_{0}|$ increases. In addition, similar behavior of $\Delta_{1,+}$ is
observed for blue detuning of the excitation.

In conventional PB, the blockade mechanism is ascribed to a reinforced vacuum
Rabi splitting, which generates strong energy-spectrum anharmonicity. When one
photon is absorbed resonantly with the higher or lower branch, one photon
excitation will ”blocks” a second photon excitation due to the larger two-
excitation energy gap, leading to an orderly output of photons one by one with
strong photon antibunching. As expected, the enhanced vacuum Rabi splitting
(energy-spectrum anharmonicity) induced by $U_{0}$ facilitates the generation
of strong PB due to a strong reduction of the spectral overlap with higher-
excitation states [Fig. 1(b)]. Recently, enhanced PB has been experimentally
demonstrated via detuning the quantum-dot and cavity resonancesMüller  _et
al._ (2015a), for which the mechanism is similar to that based on the optical
Stark shift with tuning of the vacuum Rabi splitting. However, the optical
Stark shift will further induce strong asymmetry in the higher manifolds of
the dressed states in our proposal.

##  III QUANTUM INTERFERENCE

Unlike conventional PB, which requires strong atom-cavity coupling,
unconventional PB can be achieved by utilizing quantum interference even in
the weak-coupling regime Tang  _et al._ (2015) through constructing two
different quantum transition paths between the cavity field and the classical
pumping field. In the analogous configuration of Ref. Tang  _et al._ (2015)
for a single-quantum-dot-cavity system, the present model of Eq. (II) with
application of a Raman field and microwave fields should also possess the
property that quantum interference can suppress two-photon excitation.
Therefore, strong PB will emerge when quantum interference and a large vacuum
Rabi splitting induced by an additional optical Stark shift are combined. To
proceed further, we calculate the optimal conditions for quantum interference
in the system in the following.

In the presence of a microwave field and a weak driving field, the excitation
number $n$ is not a good quantum number, which indicates that an exact
analytical solution for the system does not exist. Nevertheless, the
Hamiltonian (II) can be approximatly diagonalized in subspaces defined by a
large enough truncation excitation number of the system. Without loss of
generality, the wave function of the system is formally given by Tang  _et
al._ (2015)

$\displaystyle|\psi\rangle$
$\displaystyle=\sum_{n=0}^{\infty}C_{n,\uparrow}|n,\uparrow\rangle+\sum_{n=1}^{\infty}C_{n-1,\downarrow}|n-1,\downarrow\rangle,$
(3)

where $|C_{n,\uparrow}|^{2}$ and $|C_{n-1,\downarrow}|^{2}$ denote the
occupation probability for eigenstates $|n,\uparrow\rangle$ and
$|n-1,\downarrow\rangle$, respectively. In the single-PB regime, we need only
to truncate the two-photon ($n=2$) excitation subspaces. As a result, the wave
function for the system can be approximated as

$\displaystyle|\psi\rangle$
$\displaystyle=\sum_{\sigma=\uparrow,\downarrow}\left[C_{0,\sigma}|0,\sigma\rangle+C_{1,\sigma}|1,\sigma\rangle\right]+C_{2,\uparrow}|2,\uparrow\rangle,$
(4)

By substituting $|\psi\rangle$ into the Schrödinger equation, the probability
coefficients in the steady-state solutions are determined from the following
equations:

$\displaystyle\eta
C_{1,\uparrow}\\!+\Omega_{m}e^{i\theta}C_{0,\downarrow}\\!=0,$
$\displaystyle\eta
C_{1,\uparrow}\\!+\sqrt{2}\Delta_{1}C_{2,\uparrow}\\!+gC_{1,\downarrow}\\!=0,$
$\displaystyle\Omega_{m}e^{-i\theta}C_{1,\uparrow}\\!+\eta
C_{0,\downarrow}\\!+\sqrt{2}gC_{2,\uparrow}\\!+\Delta_{2}C_{1,\downarrow}\\!=0,$
$\displaystyle\Omega_{m}e^{-i\theta}C_{0,\uparrow}\\!+gC_{1,\uparrow}\\!+(\Delta_{c}-i\gamma_{g})C_{0,\downarrow}\\!+\eta
C_{1,\downarrow}\\!=0,$ $\displaystyle\eta
C_{0,\uparrow}\\!+\Delta_{1}C_{1,\uparrow}\\!+gC_{0,\downarrow}\\!+\sqrt{2}\eta
C_{2,\uparrow}\\!+\Omega_{m}e^{i\theta}C_{1,\downarrow}\\!=\\!0,$ (5)

where $\gamma_{g}$ is the loss rate of the atomic ground state, and
$\Delta_{1}=\Delta_{c}+U_{0}-i\kappa$ and
$\Delta_{2}=2\Delta_{c}-i\kappa-i\gamma_{g}$ are introduced as shorthand
notation.

Figure 2: Quantum transition paths in the atom-cavity system (II). The two-
photon excited state $|2,\uparrow\rangle$ (dashed line) is eliminated by
employing destructive quantum interference.

For the single-photon-excitation limit, multiphoton excitation with $n\geq 2$
must be suppressed. At least a two-photon excitation probability
$C_{2,\uparrow}=0$ is required first. For a nonzero microwave-field coupling,
the nontrivial solution of Eqs. (5) is given by

$\displaystyle\theta_{\rm{opt}}$
$\displaystyle={\arctan}(\frac{\kappa+\gamma_{g}}{2\Delta_{c}}),$
$\displaystyle\Omega_{m,{\rm opt}}$
$\displaystyle=\frac{{\eta}(\mathcal{R}+\sqrt{\mathcal{R}^{2}+4})}{2},$ (6)

where ${\mathcal{R}}={\sqrt{4{\Delta^{2}_{c}}+{(\kappa+\gamma_{g})}^{2}}}/{g}$
and $0\leq\theta_{\rm{opt}}/\pi\leq 1$. The solutions of Eq. (III) for the
optimal phase $\theta_{\rm opt}$ and Rabi frequency $\Omega_{m,{\rm opt}}$
denote the conditions for eliminating two-photon excitation by quantum
interference. To generate a large cavity output, we must focus on single-
photon resonance excitation with a fixed $\Delta_{c}=-\Delta_{1,\pm}$ in Eq.
(III). Therefore, the optimal parameters are significantly dependent on the
optical Stark shift $U_{0}$.

The essential mechanism for generating $C_{2,\uparrow}=0$ is ascribed to
destructive quantum interference between different quantum transition paths
even for a weak coupling strength of the system, see Fig. 2. This interference
can occur, for example, between the transitions
$\left|1,\uparrow\right\rangle\xrightarrow{\sqrt{2}\eta}\left|2,\uparrow\right\rangle$
induced by the driving field and
$\left|1,\uparrow\right\rangle\xrightarrow{\Omega_{m}e^{i\theta}}\left|1,\downarrow\right\rangle\xrightarrow{\sqrt{2}g}\left|2,\uparrow\right\rangle$
induced by the combination of both the microwave field and the cavity field,
after the state $|1,\uparrow\rangle$ is excited from the initial state
$|0,\uparrow\rangle$ by the driving field. Then the probability of two-photon
excitation is zero when the optimal conditions of Eq. (III) are satisfied. We
should note that the quantum interference mechanism ($C_{2,\uparrow}=0$)
vanishes in the absence of a microwave field ($\Omega_{m}=0$).

##  IV NUMERICAL RESULTS

To further demonstrate PB, we calculate the quantum statistical properties of
the system by solving the quantum master equation using the Quantum Optics
Toolbox Tan (1999). Including the dissipation and pure dephasing of the
system, the Liouvillian superoperator ${\cal{L}}$ of the atom-cavity coupled
system satisfies

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

where $\rho$ is the density matrix of the atom-cavity system, $\gamma_{d}$ is
the pure dephasing, $\sigma_{-}=|\uparrow\rangle\langle\downarrow|$ is the
lowering spin-$1/2$ operator, and
$\mathcal{D}[\hat{o}]\rho=2\hat{o}{\rho}\hat{o}^{\dagger}-\hat{o}^{\dagger}\hat{o}{\rho}-{\rho}\hat{o}^{\dagger}\hat{o}$
denotes the Lindblad type of dissipation. In experiments, the second-order
correlation function is a key physical quantity for PB, and is defined as
Glauber (1963)

$\displaystyle
g^{(2)}(\tau)=\frac{<\hat{a}^{\dagger}(t)\hat{a}^{\dagger}(t+\tau)\hat{a}(t+\tau)\hat{a}(t)>}{<\hat{a}^{\dagger}(t)\hat{a}(t)><\hat{a}^{\dagger}(t+\tau)\hat{a}(t+\tau)>}.$
(8)

For the steady-state solution with ${\cal L}\rho_{s}=0$ Tan (1999), the
second-order correlation function for zero time delay, $g^{(2)}(0)={\rm
Tr}(\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a}\rho_{s})/{n}^{2}_{s}$,
and the steady-state intracavity photon number, ${n}_{s}={\rm
Tr}(\hat{a}^{\dagger}\hat{a}\rho_{s})$, are directly calculated by solving the
quantum master equation. The nonclassical antibunching of photons is
characterized by $g^{(2)}(0)<g^{(2)}(\tau)$, and $g^{(2)}(0)=0$ is a signature
of ideal PB Mandel and Wolf (1995).

Specifically, we consider a single ultracold spin-$\frac{1}{2}$ atom in a
high-finesse optical cavity with a cavity decay rate $\kappa=2\pi\times 147$
kHz Li  _et al._ (2017). The intracavity photon number for an empty cavity is
$n_{0}=(\eta/\kappa)^{2}=0.02$ Birnbaum  _et al._ (2005), the atomic decay
rate for the ground state is $\gamma_{g}/\kappa=10^{-3}$ and the pure
dephasing is set to $\gamma_{d}/\gamma_{g}=1.0$. Taking the cavity decay rate
$\kappa$ as the energy unit, the theoretically tunable parameters include the
atom-cavity (microwave) coupling strength $g$ ($\Omega_{m}$), the phase
difference $\theta$, the optical Stark shift $U_{0}$, and the cavity-light
detuning $\Delta_{c}$. Note that $\Omega_{m}$ and $\theta$ can be fixed by
applying the quantum interference conditions of Eq. (III). In the following,
we focus only on the red-sideband excitation in Eq. (III). In fact, we have
checked that the results for blue-sideband excitation display similar
behavior. As a result, the free parameters in our system are reduced to $g$,
$U_{0}$, and $\Delta_{c}$.

Figure 3: (a) Second-order correlation function and (b) intracavity photon
number as a function of cavity-light detuning $\Delta_{c}$ and optical Stark
shift $U_{0}$ for the optimal parameters $\theta_{\rm{opt}}$ and
$\Omega_{m,\rm{opt}}$ with $g/\kappa=1$. The red dashed line shows the vacuum
Rabi splitting ($|\Delta_{1,-}|$) of the system in the red sideband. The
colors with blue-red gradient shading indicate the values of log[$g^{(2)}(0)$]
in (a) and ${n}_{s}$ in (b).

Figure 3(a) displays our numerical results for $g^{(2)}(0)$ in the
$\Delta_{c}$-$U_{0}$ parameter plane under the optimal quantum interference
conditions of Eq. (III) with $g/\kappa=1$. The red dashed line shows the red-
sideband vacuum Rabi splitting. As expected, $g^{(2)}(0)$ obviously depends on
the sign of the optical Stark shift $U_{0}$ and the cavity-light detuning
$\Delta_{c}$; the asymmetric features and the optimal $g^{(2)}(0)$ [the blue
regime in Fig. 3(a)] are consistent with the analytic energy spectrum,
especially for negative $U_{0}$, in spite of the driving fields and the cavity
decay are being included. Fig. 3(b) shows the mean cavity photon number
$n_{s}$ with the same parameters as in Fig. 3(a). As can be seen, ${n}_{s}$
also displays an asymmetric behavior dependent on the sign of $U_{0}$. When
$\Delta_{c}$ is tuned, ${n}_{s}$ exhibits two peaks near the red and blue
sidebands of the single-photon resonance points. Interestingly, the cavity
field simultaneously hosts strong PB with $g^{(2)}(0)<0.01$ and a large cavity
output in the red sideband with $\Delta_{c}\approx\Delta_{1,-}$ (red dashed
line), which is in good agreement with the vacuum Rabi splitting in Eq. (2).

We emphasize that the strong antibunching is contributed by the use of quantum
interference for suppressing two-photon excitation and the optical-Stark-
shift-enhanced vacuum Rabi splitting with increasing energy-spectrum
anharmonicity. As shown in Fig. 2, multiphoton excitations ($n\geq 3$) also
need to be suppressed simultaneously to generate strong PB, which indicates
that the quantum interference mechanism is not sufficient to guarantee the
realization of strong PB, as we will discuss below. Nevertheless, the
multiphoton excitations are far off resonance due to the strong anharmonicity
of the energy spectrum enhanced by the optical Stark shift in our system.

Figure 4: (a) and (b) $\Delta_{c}$ dependence of $g^{(2)}(0)$ and ${n}_{s}$,
respectively, for different $U_{0}$. The squares mark the position of the
minimum values of $g_{\rm{opt}}^{(2)}(0)=\min[g^{(2)}(\Delta_{c})]$ for
$U_{0}/\kappa=-5$ (solid line), $U_{0}/\kappa=0$ (dashed line), and
$U_{0}/\kappa=5$ (dotted-dashed line). Optimal results $g_{\rm{opt}}^{(2)}(0)$
(c) and the corresponding ${n}_{s,\rm{opt}}$ (d) as a function of $U_{0}$. The
other parameter is fixed at $g/\kappa=1$.

More importantly, it is shown that strong PB ($g^{(2)}(0)$) exists in a large
parameter region [the blue regime in Fig. 3(a)], which is important for
experimental feasibility when the system parameters do not perfectly satisfy
the optimal conditions. We check that, remarkably, both $g^{(2)}(0)$ and
$n_{s}$ are insensitive to weak dephasing ($\gamma_{d}/\kappa\ll 1$) in the
ground manifold for our system. However, strong dephasing
($\gamma_{d}/\kappa\gg 1$), e.g., for excited states, will induce significant
decoherence in the quantum properties of emitter-cavity coupled
systemsAuffèves  _et al._ (2010); Majumdar  _et al._ (2010, 2011, 2012b);
Englund  _et al._ (2010, 2012).

To gain more insight into PB, we plot the $\Delta_{c}$ dependence of
$g^{(2)}(0)$ and ${n}_{s}$ for different $U_{0}$ as shown in Fig. 4(a) and
(b), respectively. The optical-Stark-shift-mediated red-blue-detuning
asymmetry is observed for both $g^{(2)}(0)$ and ${n}_{s}$. In the absence of
$U_{0}$, an asymmetric structure also appeares, which is ascribed to the
quantum interference conditions in the red sideband of the system and has been
discussed in detail in our previous work Tang  _et al._ (2015). In contrast to
positive $U_{0}$, which suppresses PB, a negative $U_{0}$ can significantly
enhance PB due to the asymmetric vacuum Rabi splitting of the first manifold
in the energy spectrum, where the $U_{0}$ enhanced anharmonicity of the energy
spectrum is conducive to suppressing multiphoton excitations.

Figure 4(c) and 4(d) display the minimum value of
$g_{\rm{opt}}^{(2)}(0)=\min[g^{(2)}(\Delta_{c})]$ and the corresponding
${n}_{s,\rm{opt}}$ at the same value of the cavity-light detuning as a
function of $U_{0}$. As $U_{0}$ changes from positive to negative,
$g_{\rm{opt}}^{(2)}(0)$ decrease approximately linearly. However, the value of
${n}_{s,\rm{opt}}$ displays a peak structure which reaches a maximum around
$U_{0}=0$. Compared with the case of negative $U_{0}$, ${n}_{s,\rm{opt}}$ has
a rapid decrease with increasing positive $U_{0}$. Another typical feature is
that ${n}_{s,\rm{opt}}$ in the PB regime remains much larger than the empty-
cavity photon number when $U_{0}<0$. The largely enhanced cavity photon number
can be readily understood as being due to the additional microwave field.
Besides the weak drivening field, the additional cavity field emission
($|0,\downarrow\rangle\leftrightarrow|1,\uparrow\rangle$ transition)
originates from Bragg scattering Deng  _et al._ (2014) when the microwave
field couples to the $|0,\uparrow\rangle\leftrightarrow|1,\downarrow\rangle$
transition, as shown in Fig. 2. Our proposal for realizing a larger cavity
output while exhibiting strong antibunching is very important for applications
to high-quality single-photon sources Birnbaum  _et al._ (2005). Moreover, the
$U_{0}$ dependence of the photon quantum statistics can be realized by a
nondestructive diagnosis of the magic wavelength (zero Stark shift), which
could facilitate the study of clock transitions in ultracold atomic and ionic
gases Brusch  _et al._ (2006); Barber  _et al._ (2008); Goban  _et al._
(2012); Liu  _et al._ (2015).

Figure 5: (a) $g$ dependence of second-order correlation function
$g_{\rm{opt}}^{(2)}(0)$. (b) Dependence of $g_{\rm{opt}}^{(2)}(\tau)$ on the
time interval $\tau$ with fixed $g/\kappa=1$. The curves labeled “JC” denote
the results from the JC model, while “QI” denotes the results from our model
calculated for the optimal quantum interference conditions for $U_{0}/g=-5$.
Figure 6: Dynamic evolution of the photon number as a function of time $t$
with $g/\kappa=1$. From the first to the third row, the Stark shift is
$U_{0}/\kappa=0$, $-2$, and $-4$.

We now turn to the validity of the quantum interference mechanism for
achieving strong PB in the coupled system. In the absence of the microwave
field and the optical Stark shift, the Hamiltonian in Eq. (II) in reduces to
the paradigm of the Jaynes-Cummings (JC) Hamiltonian of a two-level atom
coupled to a cavity (harmonic oscillator), where strong PB requires a strong
atom-cavity coupling $g$ to suppress two-photon excitation Birnbaum  _et al._
(2005). Figure 5(a) shows the minimum values of $g_{\rm{opt}}^{(2)}(0)$ for
the JC model (dashed line) and our model (solid line) at $U_{0}/g=-5$ as a
function of the atom-cavity coupling $g$. In contrast to the JC model, our
$g_{\rm{opt}}^{(2)}(0)$ decreases rapidly when powerful quantum interference
conditions are utilized as $g$ increases. Interestingly, strong PB can be
easily achieved even when the system is not in the strong-coupling regime
($g/\kappa\sim 1$). However, the quantum interference mechanism is invalid for
a very weak coupling strength due to the finite cavity decay
($g/\kappa<0.38$). In this case, multiphoton ($n\geq 3$) excitations begin to
play an important role in breaking the PB because of the absence of strong
enough energy-spectrum anharmonicity, although the two-photon excitation is
eliminated by quantum interference. This explains why the PB of “QI” in Fig
5(a) is not enhanced compared with “JC” when $g/\kappa<0.38$. Interestingly,
we find that the value of the vacuum Rabi splitting $|\Delta_{1,-}|/\kappa$ is
equal to 1.97 at $g/\kappa=0.38$, which is equal to the linewidth of the
cavity ($2\kappa$). As can be seen, quantum interference can significantly
enhance the PB when $|\Delta_{1,-}|/2\kappa>1$ with $g/\kappa>0.38$, which
demonstrates that energy-spectrum anharmonicity and quantum interference are
two important factors for generating strong PB.

To further confirm the photon statistics, the time-dependent second-order
correlation function $g_{\rm{opt}}^{(2)}(\tau)$ is calculated [Fig. 5(b)]. As
can be seen, the transmitted photons allow both strong PB, with
$g_{\rm{opt}}^{(2)}(0)=0.01$, and photon antibunching, with
$g_{\rm{opt}}^{(2)}(0)<g_{\rm{opt}}^{(2)}(\tau)$. More importantly,
$g_{\rm{opt}}^{(2)}(\tau)$ increases to unity at a time $\tau\approx
1/(\kappa+\gamma_{e})=1.1$ $\mu$s, which indicates that the nonclassical
antibunching effect in the emitted photons possesses a very long coherence
time. This lifetime is much larger than that engineered with a single two-
level atom (with one ground state and one excited state) trapped in an optical
cavity, with a value of around a few tens of nanoseconds Birnbaum  _et al._
(2005), which is ascribed to the smaller single-particle loss of the atomic
ground state than that of the excited state.

Finally, to illustrate the versatility of the optical Stark shift in our
proposal, we plot the time-evolution of the photon number for different values
of $U_{0}$ in Figs. 6(a)-6(c). We find that the photon number exhibits a
quasiperiodic oscillation, with a gradually decreasing amplitude of the
oscillation with increasing time. The oscillation behavior of the photon
number depicts the quantum interference effect that is used for generating
strong PB Liew and Savona (2010); Majumdar  _et al._ (2012a); Vaneph  _et al._
(2018). As we can see, the oscillation frequency is very sensitive to the
value of $U_{0}$, which is determined by the Rabi frequency of the system.
Interestingly, the photon number rapidly reaches a saturation value for
$U_{0}=0$, as shown in Fig. 6(a). In addition, it is clear that a large
negative $U_{0}$ corresponds to a large oscillation frequency due to the
enhanced vacuum Rabi frequency [Figs. 6(b) and 6(c)], which can be used to
further enhance the understanding of the PB mechanism in our model.

##  V EXPERIMENTAL FEASIBILITY

Our proposal can be readily applied to different single alkali-metal atoms in
an optical cavity Kampschulte  _et al._ (2010); Mücke  _et al._ (2010). Taking
87Rb atom as a prototype, we now discuss in detail how to realize our scheme
with ultracold $F=1$ ground states in an experiment. An important ingredient
is the realization of a pseudospin-$\frac{1}{2}$ model with the choice
$|\uparrow\rangle=|F=1,m_{F}=0\rangle$ and
$|\downarrow\rangle=|F=1,m_{F}=1\rangle$, where the two desired Zeeman spin
states are decoupled from the remaining Zeeman spin state of the $F=1$
manifold owing to the large quadratic Zeeman shift induced by a bias magnetic
field Lin  _et al._ (2011). The cavity-mediated Raman coupling could be
generated by utilizing the experimental realization of electromagnetically
induced transparency with a single atom inside an optical cavity Kampschulte
_et al._ (2010); Mücke  _et al._ (2010). As to the cavity, the single-photon
coupling strength and cavity decay rate may be taken as $g_{0}=2\pi\times 4.5$
MHz and $\kappa=2\pi\times 0.15$ MHz, respectively, where the typical decay
rate of an excited atom, $\gamma_{e}=2\pi\times 3.0$ MHz, is much larger than
$\kappa$ Mücke  _et al._ (2010); Léonard  _et al._ (2017a). The optical Stark
shift can be tuned by changing the strength or sign of a large atom-pump
detuning, which is a demonstrated capability in current experiments on self-
organized superradiance and supersolid phases in an optical cavity Mücke  _et
al._ (2010); Mottl  _et al._ (2012); Léonard  _et al._ (2017a, b). On the
other hand, the optical Stark shift $|U_{0}|$ is naturally enhanced by further
increasing the single-photon coupling strength $g_{0}$. Importantly, all
parameters for generating strong PB in our model are highly controllable in
currently available experimental systems. As to the experimental measurement,
the cavity transmission can be collected by a single-photon-counting module
Kampschulte  _et al._ (2010), and the photon quantum statistics can be
measured by a Hanbury Brown and Twiss interferometer Birnbaum  _et al._
(2005).

##  VI CONCLUSION AND DISCUSSION

In conclusion, we propose a simple experimental scheme to realize strong PB in
a single-atom-cavity system by combining a strong vacuum Rabi splitting
induced by the optical Stark shift and quantum interference under optimal
conditions. We show that the second-order correlation function of the system
can be reduced by more than several orders of magnitude compared with the JC
model even at a moderate atom-cavity coupling, and, simultaneously, a large
cavity photon output can be obtained. As to experimental feasibility, our
proposal for generating strong PB can be applied over a fairly large parameter
region by tuning the optical Stark shift and the microwave field in a highly
controllable way. In particular, in contrast to conventional solid quantum
devices, the long coherence time based on the ground-state atom in the system
is also extremely conducive to quantum storage and detectors at the single-
photon level Witthaut  _et al._ (2012); Fleischhauer  _et al._ (2005); Chang
_et al._ (2014). Furthermore, the scheme for realizing quantum interference
and strong vacuum Rabi splitting beyond the strong-coupling limit could also
be applied to explore high-quality multiphoton blockade, e.g., photon-pair
sources could be obtained by applying destructive quantum interference to a
three-photon excitation Chang  _et al._ (2016). With slight modifications, our
proposal can be extended to various systems, including a single-quantum-dot-
cavity model Müller  _et al._ (2015b); Calic  _et al._ (2011) and a singe-ion-
cavity model Steiner  _et al._ (2013); Zhong  _et al._ (2018); Lee  _et al._
(2019), which provide several interesting opportunities for exploring the
fundamentals of quantum optics, and potential applications in single-photon
transistors, all-optical switching, and quantum metrology Faraon  _et al._
(2008); Volz  _et al._ (2012); Muñoz  _et al._ (2014).

## ACKNOWLEDGMENT

J.T. is supported by the NSFC (Grants No. 11804409) and Fundamental Research
Funds for the Central Universities (Grant No. 18lgpy80). Y.D. is supported by
the NSFC (Grant No. 11874433) and by National Key R$\&$D Program of China
(2018YFA0307500). C.H.L. is supported by the NSFC (Grants No. 11874434, No.
11574405).

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