# Entanglement of three cavity fields via resonant interactions with dressed
three-level atoms

Jinhua ZouAuthor for correspondence<EMAIL_ADDRESS>College of
Physical Science and Technology, Yangtze University, Jingzhou, 434023, China

###### Abstract

In this paper we show that three cavity fields can be entangled when they are
tuned on resonance with an ensemble of dressed three-level atoms. The master
equation for the three cavity modes is derived by using atomic dressed states
and the inseparability of the three output cavity modes is described by using
a sufficient criterion proposed by van Loock and Furusawa. The physical cause
is the atomic coherence effects, by which the quantum correlations are created
in the field dynamics.

Keywords: continuous-variable entanglement, output tripartite entanglement,
atomic coherence

###### pacs:

PACS numbers: 42.50.Dv, 03.67.Mn

##  I Introduction

Atomic coherence lies in the center of many novel effects in quantum optics
and laser physics. Electromagnetically induced transparency [1,2], coherent
population trapping [2], Hanel-effect laser [3] and quantum beat laser [4] are
such examples. Besides these, the correlation between the photons can also be
induced by atomic coherence [5-12]. One such example is the generation of
squeezed light in a three-level cascade laser using atomic coherence [5-8].
The atomic coherence can be created by preparing the atoms initially in a
coherent superposition state of the two states which are dipole-forbidden
[5-8] or driving the two states by a strong coherent field [9-11] or Raman
coupling the two states through the third auxiliary atomic states [12]. For
two-photon correlated-spontaneous-emission laser with injected atomic
coherence, it exhibits complete spontaneous-emission noise quenching and phase
squeezing simultaneously [5]. It has also been pointed out that atomic
coherence in a two-photon correlated emission laser system can be used to
generate a macroscopic two-mode entangled state and this system can be treated
as an entanglement amplifier [12].

Recently, the topic of continuous-variable entanglement has attracted much
attention as it is the base of all branches of quantum information and
communication protocols [13]. Among various entanglement generation schemes,
entanglement induced by atomic coherence has been extensively researched
[14-16]. For a nondegenerate three-level cascade laser with a subthreshold
nondegenerate parametric oscillator coupled to a vacuum reservoir, the
entanglement and squeezing for the two cavity modes in this combined system is
induced by the injected atomic coherence [14]. In a two-mode single-atom laser
with the atomic coherence exhibited by two classical laser fields,
entanglement between two field modes is demonstrated [15]. Later, it was shown
that in a three-level $\Lambda$ or V atomic system with two classical driving
fields and two cavity modes coupling corresponding transitions, by exploring
the two-channel interaction mechanism and using the squeeze-transformed modes,
continuous-variable entanglement between the two modes is obtained and the
best achievable entangled state approaches the original EPR state [16]. The
above work has mainly been confined to two-partite systems.

With the progress in continuous-variable entanglement, the generation of more
than two partite entanglement has been paid much attention as it may be the
key ingredient for advanced multiparty quantum communication such as quantum
teleportation network [17], telecloning [18] and controlled dense coding [19].
Among various generation schemes for tripartite systems, few work has been
done to generate tripartite entanglement using atomic coherence. Most
recently, a scheme to generate three-mode-entangled light fields via the
interaction between the four-level atoms and the cavity has been proposed
[20]. Three cavity modes are generated through three successive transitions in
the four-level cascade atoms. In addition to the cavity modes, two strong
classical fields drive a pair of two-photon transitions in the four-level
atoms. They show that the entanglement could only be obtained in a short time
as all the mean photons are amplified as time elapses. Thus at steady time,
the entanglement does not exist.

In this paper, we present a scheme to generate tripartite entanglement for
three cavity modes via the interaction for the three-level lambda atoms with
the three cavity modes and two classical fields. As the classical fields are
strong, the effective interaction is resonant interaction in the dressed-state
picture. We deduce the master equation of the three cavity modes by means of
the atomic dressed states and linear theory. The sufficient inseparability
criterion for continuous-variable entanglement is used to demonstrate the
entanglement properties of the three cavity modes and the results show that
our system can be used as a source to generate tripartite entangled light even
at steady state.

It is should be noted that, up to now many schemes have been proposed to
generate tripartite entanglement using linear optics or nonlinearities [17,
21-26]. It was theoretically predicted that using single-mode squeezed state
and linear optics, a truly N-partite entangled state can be generated [17].
Later, a continuous-variable tripartite entangled state was experimental
realized by combing three independent squeezed vacuum states [21]. At first,
the production of continuous-variable tripartite entanglement was presented by
mixing squeezed beams on unbalanced beamsplitters [21,22]. Recently,
generation of tripartite entanglement are focused on using cascade nonlinear
interaction in an optical cavity [23-25] or in a quasiperiodic superlattice
[26]. Among the latter are systems using parametric down-conversion with sum
frequency generation [23,25,26] or using single nonlinearity [24]. During
these nonlinear processes, the cavity modes couple with each other directly.
As these nonlinear processes are related to the higher-order polarization, the
efficiency of these processes are relatively small compared with the processes
related to linear polarization. In this way, these nonlinear processes are not
the best choice for the generation of high efficiency tripartite entangled
states.

Compared with the schemes based on the nonlinear processes [23-26], our scheme
is more effective as the generation process is resonant interaction in the
dressed states and it is only related to linear polarization. What’s more, the
linear process provides much more parameters to choose than that of the
nonlinear processes, as the atomic parameters can be varied. Compared with the
scheme in Ref. [20], our scheme can provide steady state tripartite
entanglement while the entanglement produced in scheme [20] is just kept in a
quite limited time. And in Ref. [20], they use four-level cascade atomic
system and the effective processes in the dressed states of the driving fields
are all two-photon transitions. High excited states are involved in their
scheme. In our scheme we use three-level $\Lambda$ atomic system, and the
effective processes in the dressed states of the driving fields are all
single-photon transitions. When we take into the account of the atomic
spontaneous emission, their schemes seems to have more obstacles than ours.

The paper is organized as follows. In Sec. II, we discuss the essential
ingredients of the model and deduce the density-matrix equation for the cavity
fields in a dressed-state picture. In Sec. III, we present the output
correlation spectra by solving the equations of the cavity fields and analyze
the output tripartite continuous-variable entanglement characteristics by
using a sufficient criterion proposed by van Loock and Furusawa. In Sec. IV,
we give a brief conclusion.

##  II Model and equation

We consider $N$ three-level lambda-type atoms in a three-mode cavity as shown
in Fig. 1(a). Two laser fields of frequencies $\omega_{l1,l2}$ drive the
transitions $|1,2\rangle\leftrightarrow$ $|3\rangle$, respectively. Two cavity
modes $a_{1,2}$ of frequencies $\omega_{c1,c2}$ couple the atomic transition
$|1\rangle\leftrightarrow$ $|3\rangle$, while the cavity mode $a_{3}$ with
frequency $\omega_{c3}$ couples the transition $|2\rangle\leftrightarrow$
$|3\rangle$. $\gamma_{l}$ ($l=1,2$) are the atomic decay rates from level
$|3\rangle$ to levels $|1,2\rangle$ and $\kappa_{l}$ ($l=1,2,3$) are the
cavity loss rates. For simplicity, we assume that
$\gamma_{1}=\gamma_{2}=\gamma$ and $\kappa_{1}=\kappa_{2}=\kappa_{3}=\kappa$.
The three cavity modes are assumed to be in their vacuum state initially. In
the frame of the frequencies of the laser fields and under the dipole and the
rotating-wave approximations, the total Hamiltonian is

Figure 1: (a) Atomic energy level scheme and the coupling of the cavity fields
and the classic fields. (b) Equivalent resonant transitions in the picture
dressed by the classical fields.

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{1}+H_{2}+H_{3},$
$\displaystyle H_{1}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{3}\hbar\delta_{j}a_{j}^{\dagger}a_{j},$
$\displaystyle H_{2}$ $\displaystyle=$
$\displaystyle-\hbar\Delta\left(\sigma_{11}-\sigma_{22}\right)+\hbar\Omega\left(\sigma_{31}+\sigma_{32}+H.c.\right),$
(1) $\displaystyle H_{3}$ $\displaystyle=$ $\displaystyle i\hbar
g\left(a_{1}\sigma_{31}+a_{2}\sigma_{31}+a_{3}\sigma_{32}\right)+H.c.,$

H.c. symbols the Hermitian conjugate. $H_{1}$ denotes the free energy for
three cavity fields, $H_{2}$ describes the interaction of the laser fields
with the atoms, and $H_{3}$ indicates the interaction of the cavity fields
with the atoms. $\sigma_{jk}=|j\rangle\langle k|$ ($j,k=1,2,3$) are atomic
dipole operators for $j\neq k$ and atomic projection operators for $j=k$. The
cavity detunings are defined as $\delta_{j}=\omega_{cj}-\omega_{l1}$
($j=1,2$), and $\delta_{3}=\omega_{c3}-\omega_{l2}$. The detunings of the
laser fields are defined as $\Delta_{j}=\omega_{3j}-\omega_{lj}$ ($j=1,2$),
where $\omega_{31}$ and $\omega_{32}$ are the resonance frequencies of
transitions $|1,2\rangle\leftrightarrow$ $|3\rangle$. We have assumed equal
coupling coefficients $g$ for three cavity modes, equal Rabi frequency
$\Omega$ for the two laser fields, and opposite detunings of the two laser
fields $\Delta_{1}=-\Delta_{2}=\Delta$.

We assume that the laser fields are much stronger than the cavity fields,
i.e., $\Omega\gg g\langle a_{l}\rangle$, ($l=1,2,3)$. The laser fields can be
viewed as dressing fields for the atoms. Therefore, by diagonalizing the
Hamiltonian $H_{2}$, we find the so-called semiclassical dressed states as

$\displaystyle|0\rangle$ $\displaystyle=$
$\displaystyle-\frac{c}{\sqrt{2}}|1\rangle+\frac{c}{\sqrt{2}}|2\rangle+s|3\rangle,$
$\displaystyle|+\rangle$ $\displaystyle=$
$\displaystyle\frac{1+s}{2}|1\rangle+\frac{1-s}{2}|2\rangle+\frac{c}{\sqrt{2}}|3\rangle,$
(2) $\displaystyle|-\rangle$ $\displaystyle=$
$\displaystyle\frac{1-s}{2}|1\rangle+\frac{1+s}{2}|2\rangle-\frac{c}{\sqrt{2}}|3\rangle,$

where $c=\frac{\sqrt{2}\Omega}{d}$, $s=-\frac{\Delta}{d}$, and
$d=\sqrt{\Delta^{2}+2\Omega^{2}}$.

Now, we use the Hamiltonian $H_{0}=\hbar
d\left(\sigma_{++}-\sigma_{--}\right)+H_{1}$ to perform the unitary dressing
transformation. By choosing the cavity detunings as
$\delta_{1}=d=-\delta_{2}=-\delta_{3}$, and neglecting the fast-oscillating
terms such as $e^{\pm i2dt}$, we obtain the resonant interaction Hamiltonian
as

$V=ig\hbar\left(c_{3}a_{1}^{\dagger}+c_{2}a_{2}+c_{1}a_{3}\right)\sigma_{0+}+ig\hbar\left(c_{1}a_{1}-c_{3}a_{2}^{\dagger}+c_{3}a_{3}^{\dagger}\right)\sigma_{0-}+H.c.,$
(3)

where $c_{1}=\frac{1}{2}s(1-s)$, $c_{2}=\frac{1}{2}s(1+s)$, and
$c_{3}=\frac{1}{2}c^{2}$. The resonant transitions in the dressed states are
shown in Fig. 1(b).

The master equation for the cavity modes is obtained by using the usual
approach [2], starting from
$\frac{d}{dt}\rho=-\frac{i}{\hbar}\left[V,\rho\right]{\cal+L}_{a}\rho+{\cal
L}_{c}\rho$, where ${\cal
L}_{c}\rho=\frac{{}_{\kappa}}{2}\sum_{l=1}^{3}\left(2a_{l}\rho
a_{l}^{\dagger}-a_{l}^{\dagger}a_{l}\rho-\rho a_{l}^{\dagger}a_{l}\right)$ and
${\cal L}_{a}\rho$ describes the atomic decay in the dressed states picture
and its expression is very complicated. The detailed form of atomic decay term
${\cal L}_{a}\rho$ is given in Appendix A. The master equation for the cavity
modes is obtained by tracing out the atomic states, which gives
$\frac{d}{dt}\rho_{c}=g\left(c_{3}a_{1}^{\dagger}+c_{2}a_{2}+c_{1}a_{3}\right)\rho_{+0}+g\left(c_{1}a_{1}-c_{3}a_{2}^{\dagger}+c_{3}a_{3}^{\dagger}\right)\rho_{-0}+H.c$,
where $\rho_{jk}=tr_{atom}(\sigma_{kj}\rho)$ ($j,k=0,+,-$). As the atomic
variables vary much faster than the cavity fields, it is possible to express
$\rho_{jk}=tr_{atom}(\sigma_{kj}\rho)$ ($jk=+0,-0,0+,0-$) in terms of
$\rho_{c}$, $a_{l}$ and $a_{l}^{\dagger}$ ($l=$1-3) from the quasi-steady-
state solution of the coupled equations for
$\rho_{jk}=tr_{atom}(\sigma_{kj}\rho)$ ($jk=+0,-0,0+,0-$). By using
$\rho_{jj}\simeq\rho_{jj}^{s}\rho_{c}$ ($j=0,+,-$) and $\rho_{+-}^{s}\simeq
0$, where “s” implies the steady-state solutions of the density matrix
equations in the dressed state picture without the quantum fields $a_{l}$ and
$a_{l}^{\dagger}$ ($l=$1-3). The steady state populations is obtained as
$\rho_{00}^{s}=\frac{c^{4}}{1+3s^{4}}$ and
$\rho_{++}^{s}=\rho_{--}^{s}=\frac{s^{2}\left(1+s^{2}\right)}{1+3s^{4}}$. The
master equation for the cavity modes is obtained as

$\displaystyle\frac{d}{dt}\rho_{c}$ $\displaystyle=$
$\displaystyle\sum_{l=1}^{3}\left\\{A_{ll}\left[a_{l}^{\dagger},\rho_{c}a_{l}\right]-\left(B_{ll}+\frac{\kappa_{l}}{2}\right)\left[a_{l}^{\dagger},a_{l}\rho_{c}\right]\right\\}$
$\displaystyle+\sum_{l=2}^{3}\left\\{A_{1l}\left[a_{1}^{\dagger},\rho_{c}a_{l}^{\dagger}\right]-B_{1l}\left[a_{1}^{\dagger},a_{l}^{\dagger}\rho_{c}\right]+A_{l1}\left[a_{l}^{\dagger},\rho_{c}a_{l}^{\dagger}\right]-B_{l1}\left[a_{l}^{\dagger},a_{1}^{\dagger}\rho_{c}\right]\right\\}$
$\displaystyle+\sum_{l,k=2;l\neq
k}^{3}\left\\{A_{lk}\left[a_{l}^{\dagger},\rho_{c}a_{k}\right]-B_{lk}\left[a_{l}^{\dagger},a_{k}\rho_{c}\right]\right\\}+H.c..$

The explicit expressions for $A_{lk}$ and $B_{lk}$ ($l,k=1,2,3$) are given in
Appendix B. Here the terms $A_{ll}$ ($l$=1-3) and $B_{ll}$ ($l$=1-3) represent
the gain term and the absorption of mode $a_{l}$, respectively. And the terms
$A_{lk}$ and $B_{lk}$ ($l\neq k$) represent the coupling between the two modes
$a_{l}$ and $a_{k}$, and we will show that these quantities are responsible
for entanglement among three cavity fields. It is easy to see that without
these coupling terms between different cavity fields, the quantum correlation
can not be introduced among the three cavity modes. Thus entanglement among
the three cavity fields is attributed to the atomic coherence created through
the interaction between the fields and the atoms.

##  III Correlation spectra

The master equation (4) enables us to derive equations of motion for the
cavity modes:

$\displaystyle\tau\frac{d}{dt}a_{1}^{\dagger}$ $\displaystyle=$
$\displaystyle\left(A_{11}-B_{11}-\frac{\kappa_{1}}{2}\right)a_{1}^{\dagger}+\left(A_{12}-B_{12}\right)a_{2}+\left(A_{13}-B_{13}\right)a_{3}+\sqrt{\kappa_{1}}a_{1}^{\dagger
in},$ $\displaystyle\tau\frac{d}{dt}a_{2}$ $\displaystyle=$
$\displaystyle\left(A_{21}-B_{21}\right)a_{1}^{\dagger}+\left(A_{22}-B_{22}-\frac{\kappa_{2}}{2}\right)a_{2}+\left(A_{23}-B_{23}\right)a_{3}+\sqrt{\kappa_{2}}a_{2}^{in},$
(5) $\displaystyle\tau\frac{d}{dt}a_{3}$ $\displaystyle=$
$\displaystyle\left(A_{31}-B_{31}\right)a_{1}^{\dagger}+\left(A_{32}-B_{32}\right)a_{2}+\left(A_{33}-B_{33}-\frac{\kappa_{3}}{2}\right)a_{3}+\sqrt{\kappa_{3}}a_{3}^{in},$

where $\tau$ is the round-trip time of light in the cavity and assumed to be
the same for three cavity modes. $a_{j}^{in}$ and $a_{j}^{\dagger in}$
($j=$1-3) are annihilation and creation operators of the input fields to the
cavity. This is a set of linear equations. In order to solve this equation, we
use the Fourier transformation and the boundary conditions at the mirror
between the output quantities and the input quantities
$a_{j}^{in}+a_{j}^{out}=\sqrt{\kappa_{j}}a_{j}$ ($j=$1-3) to obtain the
equation in the frequency domain as

$a^{out}\left(\omega\right)=-\left(I+BD_{0}^{-1}B\right)a^{in}\left(\omega\right),$
(6)

where

$a^{out}\left(\omega\right)=\left(a_{1}^{\dagger
out}\left(-\omega\right),a_{2}^{out}\left(\omega\right),a_{3}^{out}\left(\omega\right)\right)^{T},$
---
$a^{in}\left(\omega\right)=\left(a_{1}^{\dagger
in}\left(-\omega\right),a_{2}^{in}\left(\omega\right),a_{3}^{in}\left(\omega\right)\right)^{T},$
(7)
$D_{0}=\left(\begin{array}[]{ccc}A_{11}-B_{11}-\frac{\kappa_{1}}{2}-i\omega\tau&A_{12}-B_{12}&A_{13}-B_{13}\\\
A_{21}-B_{21}&A_{22}-B_{22}-\frac{\kappa_{2}}{2}-i\omega\tau&A_{23}-B_{23}\\\
A_{31}-B_{31}&A_{32}-B_{32}&A_{33}-B_{33}-\frac{\kappa_{3}}{2}-i\omega\tau\end{array}\right),$
(8) $B=\left(\begin{array}[]{lll}\sqrt{\kappa_{1}}&0&0\\\
0&\sqrt{\kappa_{2}}&0\\\
0&0&\sqrt{\kappa_{3}}\end{array}\right),\text{\qquad}I=\left(\begin{array}[]{lll}1&0&0\\\
0&1&0\\\ 0&0&1\end{array}\right).$ (9)

where T symbols the matrix transpose.

Figure 2: The quantum correlations spectra
$S_{123}^{out}\left(\omega^{\prime}\right)$,
$S_{231}^{out}\left(\omega^{\prime}\right)$ and
$S_{312}^{out}\left(\omega^{\prime}\right)$ versus the normalized analyzing
frequency $\omega^{\prime}$ are plotted for (a) $\Delta=5$ and (b) $\Delta=10$
by solid, dashed and dotted line, respectively. The other parameters are
$\Omega=35$, $g^{2}N=10$, $\gamma=1$ and $\kappa=0.1$. Figure 3: The quantum
correlations spectra $S_{123}^{out}\left(\omega^{\prime}\right)$,
$S_{231}^{out}\left(\omega^{\prime}\right)$ and
$S_{312}^{out}\left(\omega^{\prime}\right)$ versus the detuning $\Delta$ for
(a) $\omega^{\prime}=0$ and (b) $\omega^{\prime}=1.0$ by solid, dashed and
dotted line, respectively. The other parameters are the same as those in Fig.
2.

In order to study the entanglement properties of output cavity modes, we need
to use quadrature amplitude and phase operators defined by

$\displaystyle X_{j}^{out}$ $\displaystyle=$ $\displaystyle
a_{j}^{out}\left(\omega\right)+a_{j}^{\dagger out}\left(-\omega\right),$
$\displaystyle Y_{j}^{out}$ $\displaystyle=$
$\displaystyle-i\left[a_{j}^{out}\left(\omega\right)-a_{j}^{\dagger
out}\left(-\omega\right)\right],$ (10)

Using Eq. (6) and $X_{j}^{in}=a_{j}^{in}\left(\omega\right)+a_{j}^{\dagger
in}\left(-\omega\right)$,
$Y_{j}^{in}=-i\left[a_{j}^{in}\left(\omega\right)+a_{j}^{\dagger
in}\left(-\omega\right)\right]$, we can obtain the relationships between the
input fields and the output fields as

(13) (16) (19)

where we have defined the normalized analyzing frequency
$\omega^{\prime}=\omega\tau/\kappa$. The explicit expressions for $D_{jk}$
($j,k$=1-3) are presented in Appendix C.

The presence of entanglement between the three cavity modes can be
investigated using the sufficient criterion for continuous-variable tripartite
system proposed by van Loock and Furusawa [27]. The sufficient inseparability
criterion for continuous variable tripartite entanglement is that if any one
of the following inequalities is satisfied, genuine tripartite entanglement is
demonstrated. The inequalities are

$\displaystyle S_{123}$ $\displaystyle=$ $\displaystyle
V\left[X_{1}+\left(X_{2}+X_{3}\right)/\sqrt{2}\right]+V\left[Y_{1}-\left(Y_{2}+Y_{3}\right)/\sqrt{2}\right]<4,$
$\displaystyle S_{231}$ $\displaystyle=$ $\displaystyle
V\left[X_{2}+\left(X_{3}+X_{1}\right)/\sqrt{2}\right]+V\left[Y_{2}-\left(Y_{3}+Y_{1}\right)/\sqrt{2}\right]<4,$
(20) $\displaystyle S_{312}$ $\displaystyle=$ $\displaystyle
V\left[X_{3}+\left(X_{1}+X_{2}\right)/\sqrt{2}\right]+V\left[Y_{3}-\left(Y_{1}+Y_{2}\right)/\sqrt{2}\right]<4,$

where $V\left(A\right)=<A^{2}>-<A>^{2}$. From the above definition, the
correlation spectra of the quadratures of three output cavity fields are
obtained as

$\displaystyle S_{123}^{out}\left(\omega^{\prime}\right)$ $\displaystyle=$
$\displaystyle|\sqrt{2}D_{11}-D_{21}-D_{31}|^{2}+|\sqrt{2}D_{12}-D_{22}-D_{32}|^{2}+|\sqrt{2}D_{13}-D_{23}-D_{33}|^{2},$
$\displaystyle S_{231}^{out}\left(\omega^{\prime}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(|\sqrt{2}D_{21}-D_{11}-D_{31}|^{2}+|\sqrt{2}D_{22}-D_{12}-D_{32}|^{2}+|\sqrt{2}D_{23}-D_{13}-D_{33}|^{2}\right)$
(21)
$\displaystyle+\frac{1}{2}\left(|\sqrt{2}D_{21}-D_{11}+D_{31}|^{2}+|\sqrt{2}D_{22}-D_{12}+D_{32}|^{2}+|\sqrt{2}D_{23}-D_{13}+D_{33}|^{2}\right),$
$\displaystyle S_{312}^{out}\left(\omega^{\prime}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(|\sqrt{2}D_{31}-D_{11}-D_{21}|^{2}+|\sqrt{2}D_{32}-D_{12}-D_{22}|^{2}+|\sqrt{2}D_{33}-D_{13}-D_{23}|^{2}\right)$
$\displaystyle+\frac{1}{2}\left(|\sqrt{2}D_{31}+D_{11}-D_{21}|^{2}+|\sqrt{2}D_{32}-D_{12}+D_{22}|^{2}+|\sqrt{2}D_{33}-D_{13}+D_{23}|^{2}\right).$

The quantum correlations spectra $S_{123}^{out}\left(\omega^{\prime}\right)$,
$S_{231}^{out}\left(\omega^{\prime}\right)$ and
$S_{312}^{out}\left(\omega^{\prime}\right)$ for three output cavity fields
described in Eq. (13) versus the normalized analyzing frequency
$\omega^{\prime}$ are plotted in Fig. $2$ for (a) $\Delta=5$ and (b)
$\Delta=10$ by solid, dashed and dotted line, respectively. The other
parameters are $\Omega=35$, $g^{2}N=10$, $\gamma=1$ and $\kappa=0.1$. The
satisfaction of one of the three inequalities
$S_{123}^{out}\left(\omega^{\prime}\right)<4$,
$S_{231}^{out}\left(\omega^{\prime}\right)<4$ and
$S_{312}^{out}\left(\omega^{\prime}\right)<4$ is sufficient to demonstrate
genuine tripartite entanglement. In order to analyze the entanglement
properties of the three cavity modes, we present all three correlations
$S_{ijk}^{out}\left(\omega^{\prime}\right)$ and find that the indices of the
three cavity modes are crucial. When the cavity modes are symmetric, the
indices of the cavity modes are not important as the three correlations give
the same result. But when the cavity modes are asymmetric, the indices are
crucial in that the three correlations will give different results. As shown
in Fig. 2(a), all three correlations are below 4 in a wide frequency range
thus all three inequalities are satisfied. So the three output cavity modes
are entangled. Among the three correlations,
$S_{123}^{out}\left(\omega^{\prime}\right)$ gives the minimum values with the
same parameters. When the inequalities are satisfied, the smaller the values
of correlations are the larger the correlation degree. When we increase the
detuning $\Delta$ to $10$ and keep other parameters unchanged as shown in Fig.
2(b), correlations $S_{123}^{out}\left(\omega^{\prime}\right)$ and
$S_{231}^{out}\left(\omega^{\prime}\right)$ are always below $4$ in a wide
frequency range while the correlation
$S_{312}^{out}\left(\omega^{\prime}\right)$ is larger than $4$ in a frequency
zone around the central analyzing frequency $\omega^{\prime}=0$. Thus
tripartite entanglement is also demonstrated between the three output cavity
modes. Compared with Fig. 2(a), the minimum value of
$S_{123}^{out}\left(\omega^{\prime}\right)$ is smaller, which means that the
correlation degree is also increased with the detuning. For both cases, we
also see that the large correlation can be obtained at low analyzing frequency
$\omega^{\prime}$.

In Fig. $3$, we plot $S_{123}^{out}\left(\omega^{\prime}\right)$,
$S_{231}^{out}\left(\omega^{\prime}\right)$ and
$S_{312}^{out}\left(\omega^{\prime}\right)$ as a function of detuning $\Delta$
for (a) $\omega^{\prime}=0$ and (b) $\omega^{\prime}=1.0$ by solid, dashed and
dotted line, respectively. The remain parameters are the same as those in Fig.
2. We also see that the correlation
$S_{123}^{out}\left(\omega^{\prime}\right)$ gives the minimum values with the
same parameters. It is seen from Fig. 3(a) and 3(b) that, correlations
$S_{123}^{out}\left(\omega^{\prime}\right)$ and
$S_{231}^{out}\left(\omega^{\prime}\right)$ always satisfy the inequalities
while $S_{312}^{out}\left(\omega^{\prime}\right)$ only satisfy the inequality
in a small frequency range. Thus tripartite entanglement between the three
output cavity modes is demonstrated again. It is worthwhile to point out that
when the analyzing frequency $\omega^{\prime}=0$, the system reaches its
steady state. Thus at steady state, we can also obtain entangled tripartite
light. This is in contrast with the results in Ref. [20], where the
entanglement between the three cavity modes is time dependent. In that case
all the mean photon numbers are amplified as time increases. Thus, the
entanglement for three cavity modes can not be kept for a long time. And among
the three correlations, $S_{123}^{out}\left(\omega^{\prime}\right)$ decreases
with the increasing detuning, while correlations
$S_{231}^{out}\left(\omega^{\prime}\right)$ and
$S_{312}^{out}\left(\omega^{\prime}\right)$ first decrease than increase with
increasing detuning. So correlation
$S_{123}^{out}\left(\omega^{\prime}\right)$ is the best choice when we
investigate the entanglement properties of the three cavity modes. Compared
with Fig. 3(a) and 3(b), we find that the minimal values of correlations in
Fig. 3(a) are smaller than those in Fig. 3(b). This indicates that the
correlation degree is large when the analyzing frequency $\omega^{\prime}$ is
small.

##  IV Conclusion

In conclusion, we have examined the entanglement properties of three cavity
modes interacting with three-level $\Lambda$ atomic system coupled by two
extra classical fields. As the classical fields are stronger than the cavity
fields, we adopt the dressed-atom approach to calculate the equation for the
cavity fields. After tracing out the atomic variables, we obtain the master
equation of the cavity modes and analyze the entanglement properties of the
output fields. The tripartite entanglement of the three output fields is
demonstrated theoretically by a sufficient inseparability criterion and the
entanglement characteristics are presented. This scheme of three-mode
continuous variable entanglement generation using atomic coherence is useful
in quantum information processing.

Acknowledgments

This work is supported by the Scientific Research Plan of the Provincial
Education Department in Hubei (Grant No. Q20101304) and NSFC under Grant No.
11147153.

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Appendix A

In this Appendix, we present the atomic decay term in terms of the dressed
atomic states as

$\displaystyle{\cal L}_{a}\rho$ $\displaystyle=$
$\displaystyle\sum_{j,k=0,+,-;j\neq k}\left({\cal L}_{jk}^{kj}\rho+{\cal
L}_{ph}^{kj}\rho\right)+\sum_{j,k=+,-;j\neq k}{\cal L}_{in}^{kj}\rho,$
$\displaystyle{\cal L}_{jk}^{kj}\rho$ $\displaystyle=$
$\displaystyle\frac{{}_{\gamma_{jk}}}{2}\left(2\sigma_{p}^{kj}\rho\sigma_{p}^{kj}-\sigma_{p}^{kj}\sigma_{p}^{kj}\rho-\rho\sigma_{p}^{kj}\sigma_{p}^{kj}\right),$
$\displaystyle{\cal L}_{ph}^{kj}\rho$ $\displaystyle=$
$\displaystyle\epsilon_{kj}\frac{{}_{\gamma_{ph}^{kj}}}{4}\left(2\sigma_{jk}\rho\sigma_{kj}-\sigma_{kj}\sigma_{jk}\rho-\rho\sigma_{kj}\sigma_{jk}\right),$
() $\displaystyle\sigma_{p}^{kj}$ $\displaystyle=$
$\displaystyle\sigma_{kk}-\sigma_{jj},$ $\displaystyle{\cal L}_{in}^{kj}\rho$
$\displaystyle=$
$\displaystyle\gamma_{c}\left(\sigma_{j0}\rho\sigma_{k0}+\sigma_{0j}\rho\sigma_{0k}\right),$

with $\epsilon_{kj}=1$, for $k,j=0+,0-,+-$, otherwise $\epsilon_{kj}=0$. The
parameters in the above equations are

$\displaystyle\gamma_{+-}$ $\displaystyle=$
$\displaystyle\gamma_{-+}=\frac{\gamma}{4}c^{2}\left(1+s^{2}\right),$
$\displaystyle\gamma_{+0}$ $\displaystyle=$
$\displaystyle\gamma_{-0}=\frac{\gamma}{2}s^{2}\left(1+s^{2}\right),$
$\displaystyle\gamma_{0+}$ $\displaystyle=$
$\displaystyle\gamma_{0-}=\frac{\gamma}{2}c^{4},\gamma_{c}=\frac{\gamma}{2}c^{2}s^{2},$
() $\displaystyle\gamma_{ph}^{0+}$ $\displaystyle=$
$\displaystyle{\cal\gamma}_{ph}^{0-}=\gamma
c^{2}s^{2},\gamma_{ph}^{+-}=\frac{\gamma}{2}c^{4}.$

Appendix B

In this appendix, we present the explicit expressions for the coefficients
$A_{jk}$ and $B_{jk}$ ($j,k=$1-3) in the equation of motion for the density
operator $\rho_{c\text{ }}$of the cavity modes (Eq. ($4$) ):

$\displaystyle A_{11}$ $\displaystyle=$ $\displaystyle
g^{2}N\left(c_{1}e_{1}\rho_{00}^{s}+c_{3}e_{2}\rho_{++}^{s}\right)\text{,
}B_{11}=g^{2}N\left(c_{3}e_{2}\rho_{00}^{s}+c_{1}e_{1}\rho_{--}^{s}\right),$
$\displaystyle A_{22}$ $\displaystyle=$ $\displaystyle
g^{2}N\left(c_{2}e_{3}\rho_{00}^{s}+c_{3}e_{4}\rho_{--}^{s}\right)\text{,
}B_{22}=g^{2}N\left(c_{2}e_{3}\rho_{++}^{s}+c_{3}e_{4}\rho_{00}^{s}\right),$
$\displaystyle A_{33}$ $\displaystyle=$ $\displaystyle
g^{2}N\left(c_{1}e_{1}\rho_{00}^{s}+c_{3}e_{2}\rho_{--}^{s}\right)\text{,
}B_{33}=g^{2}N\left(c_{1}e_{1}\tilde{\rho}_{++}+c_{3}e_{2}\rho_{00}^{s}\right),$
$\displaystyle A_{12}$ $\displaystyle=$ $\displaystyle
g^{2}N\left(c_{2}e_{2}\rho_{++}^{s}-c_{3}e_{1}\rho_{00}^{s}\right)\text{,
}B_{12}=g^{2}N\left(c_{2}e_{2}\rho_{00}^{s}-c_{3}e_{1}\rho_{--}^{s}\right),$
$\displaystyle A_{13}$ $\displaystyle=$ $\displaystyle
g^{2}N\left(c_{1}e_{2}\rho_{++}^{s}+c_{3}e_{1}\rho_{00}^{s}\right)\text{,
}B_{13}=g^{2}N\left(c_{1}e_{2}\rho_{00}^{s}+c_{3}e_{1}\rho_{--}^{s}\right),$
() $\displaystyle A_{21}$ $\displaystyle=$ $\displaystyle
g^{2}N\left(c_{3}e_{3}\rho_{00}^{s}-c_{1}e_{4}\rho_{--}^{s}\right)\text{,
}B_{21}=g^{2}N\left(c_{3}e_{3}\rho_{++}^{s}-c_{1}e_{42}\rho_{00}^{s}\right),$
$\displaystyle A_{23}$ $\displaystyle=$ $\displaystyle
g^{2}N\left(c_{1}e_{3}\rho_{00}^{s}-c_{3}e_{4}\rho_{--}^{s}\right)\text{,
}B_{23}=g^{2}N\left(c_{1}e_{3}\rho_{++}^{s}-c_{3}e_{4}\rho_{00}^{s}\right),$
$\displaystyle A_{31}$ $\displaystyle=$ $\displaystyle
g^{2}N\left(c_{3}e_{1}\rho_{00}^{s}+c_{1}e_{2}\rho_{--}^{s}\right)\text{,
}B_{31}=g^{2}N\left(c_{3}e_{1}\rho_{++}^{s}+c_{1}e_{2}\rho_{00}^{s}\right),$
$\displaystyle A_{32}$ $\displaystyle=$ $\displaystyle
g^{2}N\left(c_{2}e_{1}\rho_{00}^{s}-c_{3}e_{2}\rho_{--}^{s}\right)\text{,
}B_{32}=g^{2}N\left(c_{2}e_{1}\rho_{++}^{s}-c_{3}e_{2}\rho_{00}^{s}\right).$

where $e_{1}=\Gamma c_{1}-\gamma_{c}c_{3}$, $e_{2}=\Gamma
c_{3}-\gamma_{c}c_{1}$, $e_{3}=$ $\Gamma c_{2}+\gamma_{c}c_{3}$ and
$e_{4}=\Gamma c_{3}+\gamma_{c}c_{2}$ with
$\Gamma=\gamma_{ph}^{0+}+\frac{1}{2}\left(\gamma_{+-}+\gamma_{+0}+\gamma_{-0}+\gamma_{0+}\right)+\frac{1}{4}\left(\gamma_{ph}^{0-}+\gamma_{ph}^{+-}\right)$.

Appendix C

In this appendix, we will give the explicit expressions for the coefficients
$D_{jk}$ ($j,k=$1-3) in the relations between the input fields and the output
fields in Eq. (11):

$\displaystyle D_{11}$ $\displaystyle=$
$\displaystyle-1+\chi_{0}\left[\chi_{22}\chi_{33}-\left(A_{13}-B_{13}\right)\left(A_{32}-B_{32}\right)\right],$
$\displaystyle D_{22}$ $\displaystyle=$
$\displaystyle-1+\chi_{0}\left[\chi_{11}\chi_{33}-\left(A_{13}-B_{13}\right)\left(A_{31}-B_{31}\right)\right],$
$\displaystyle D_{33}$ $\displaystyle=$
$\displaystyle-1+\chi_{0}\left[\chi_{11}\chi_{22}-\left(A_{12}-B_{12}\right)\left(A_{21}-B_{21}\right)\right],$
$\displaystyle D_{12}$ $\displaystyle=$
$\displaystyle\chi_{0}\left[\left(A_{13}-B_{13}\right)\left(A_{32}-B_{32}\right)-\left(A_{12}-B_{12}\right)\chi_{33}\right],$
$\displaystyle D_{13}$ $\displaystyle=$
$\displaystyle\chi_{0}\left[\left(A_{12}-B_{12}\right)\left(A_{23}-B_{23}\right)-\left(A_{13}-B_{13}\right)\chi_{22}\right],$
() $\displaystyle D_{21}$ $\displaystyle=$
$\displaystyle\chi_{0}\left[\left(A_{23}-B_{23}\right)\left(A_{31}-B_{31}\right)-\left(A_{21}-B_{21}\right)\chi_{33}\right],$
$\displaystyle D_{23}$ $\displaystyle=$
$\displaystyle\chi_{0}\left[\left(A_{13}-B_{13}\right)\left(A_{21}-B_{21}\right)-\left(A_{23}-B_{23}\right)\chi_{11}\right],$
$\displaystyle D_{31}$ $\displaystyle=$
$\displaystyle\chi_{0}\left[\left(A_{32}-B_{32}\right)\left(A_{21}-B_{21}\right)-\left(A_{31}-B_{31}\right)\chi_{22}\right],$
$\displaystyle D_{32}$ $\displaystyle=$
$\displaystyle\chi_{0}\left[\left(A_{12}-B_{12}\right)\left(A_{31}-B_{31}\right)-\left(A_{32}-B_{32}\right)\chi_{11}\right],$

where

$\displaystyle|D_{0}|$ $\displaystyle=$
$\displaystyle\chi_{11}\left[\chi_{22}\chi_{33}-\left(A_{23}-B_{23}\right)\left(A_{32}-B_{32}\right)\right]$
$\displaystyle+\left(A_{12}-B_{12}\right)\left[\left(A_{23}-B_{23}\right)\left(A_{31}-B_{31}\right)-\left(A_{21}-B_{21}\right)\chi_{33}\right]$
$\displaystyle+\left(A_{13}-B_{13}\right)\left[\left(A_{21}-B_{21}\right)\left(A_{32}-B_{32}\right)-\left(A_{31}-B_{31}\right)\chi_{22}\right].$

with the parameters
$\chi_{jj}=\kappa\left(\frac{A_{jj}}{\kappa}-\frac{B_{jj}}{\kappa}-\frac{1}{2}-i\omega^{\prime}\right)$
($j=$1-3) and $\chi_{0}=-\frac{\kappa}{|D_{0}|}$. And we have used the
equation $\kappa_{1}=\kappa_{2}=\kappa_{3}=\kappa$.