Physics Letters A 372 (2008) 6456–6460
Contents lists available at ScienceDirect
Physics Letters A
www.elsevier.com/locate/pla
Effect of spontaneously generated coherence on Kerr nonlinearity
in a four-level atomic system
Xiang-an Yan a,b , Li-qiang Wang c , Bao-yin Yin a , Wen-juan Jiang a , Huai-bin Zheng a ,
Jian-ping Song a,∗ , Yan-peng Zhang a
a
b
c
Key Laboratory for Physical Electronics and Devices of the Ministry of Education, Xi’an Jiaotong University, Xi’an 710049, China
School of Science, Xi’an Polytechnic University, Xi’an 710048, China
Department of Applied Mathematics and Applied Physics, Xi’an Post and Telecommunications College, Xi’an 710061, China
a r t i c l e
i n f o
Article history:
Received 16 June 2008
Received in revised form 26 August 2008
Accepted 27 August 2008
Available online 2 September 2008
Communicated by P.R. Holland
a b s t r a c t
We propose a scheme for giant enhancement of the Kerr nonlinearity in a four-level atomic system in
which spontaneously generated coherence is present. The physics mechanism of the enhancement of Kerr
nonlinearity is mainly based on the presence of an extra atomic coherence induced by the spontaneously
generated coherence. Numerical values obtained by solving the density matrix equations agree well with
these exact analytical values.
© 2008 Elsevier B.V. All rights reserved.
PACS:
42.50.Gy
42.65.-k
42.50.Md
Keywords:
Nonlinear optics
Electromagnetically induced transparency
Quantum interference
Spontaneously generated coherence
Double dark resonance
Kerr nonlinearity
1. Introduction
Quantum coherence and interference in an atomic system have
shown a number of important phenomena, such as lasing without inversion (LWI) [1–4], cancellation of spontaneous emission
[5], subluminal and superluminal light [6–9], the enhancement of
the refractive index [10–12], coherent population trapping (CPT)
[13], the electromagnetically induced transparency (EIT) [14–17],
and so on. There are many ways to generate quantum coherence
and interference in an atomic system. Generally, they are produced
by a coherent driving field or by initial coherence. It is now well
known that another kind of coherence, spontaneously generated
coherence (SGC), can be created by the interference of spontaneous
emission channels. Such coherence can be created when a single excited state decays to a closely spaced lower doublet (-type
atom) [18,19] or a closely spaced excited doublet decays to a single
ground state (V-type atom) [20,21], and it can also be created in
*
Corresponding author.
E-mail addresses: [email protected], [email protected] (J.-p. Song).
0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.physleta.2008.08.056
the equispaced-atomic-level case (ladder-type atom) [22,23]. The
existence of such coherence effects depends on the nonorthogonality of the two dipole matrix elements. The study of SGC has
attracted considerable attention [24–38]. At present the effects of
SGC on gain, dispersion and populations in an atomic system have
been intensively investigated [19,25,39,40]. All the interesting features due to SGC could have useful applications in laser physics
and other areas of quantum optics.
Little work, however, has been reported on the effect of SGC
on the Kerr nonlinearity. The Kerr nonlinearity corresponds to the
refractive part of the third-order susceptibility in optical media.
Since optical nonlinear susceptibilities play important roles in alloptical switch, all-optical logic operation, polarization phase gates,
and so on, it is desirable to have large nonlinear susceptibilities
with small absorption under conditions of low light powers. In a
four-level double dark resonance system without SGC, Niu et al.
proposed that the interacting double-dark resonance can be effective in enhancing the Kerr nonlinearity by proper tuning of the
coherent control field [41]. The generic four-level -type system
consists of a single excited state decaying to three closely spaced
lower levels, which can be treated as two -type system, the com-
X.-a. Yan et al. / Physics Letters A 372 (2008) 6456–6460
6457
ρ̇34 = i Ωc ρ24 + i Ωd (ρ44 − ρ33 ).
Fig. 1. Generic four-level -type system with a twofold level structure.
mon excited state is coupled by the same vacuum modes to two
lower closely spaced levels leading to spontaneously generated coherence effect. So, in this Letter, we explore the main effect of
SGC on the Kerr nonlinearity in the same four-level atomic system with double dark resonances. We find that the enhancement
of Kerr nonlinearity can be obtained even without the condition of
proper detuning of the coherent control field. Moreover, in the case
of optimal SGC, a giant Kerr nonlinearity accompanied by vanishing absorption can be realized near zero probe detuning. We also
present the exact analytical explain, which is agreed well with the
numerical results obtained by solving the density matrix equations.
The Letter is organized as follows. In Section 2 we describe the
model and present the density-matrix equations of motion for the
system. In Section 3 we discuss the effects of SGC on the Kerr
nonlinearity and nonlinear absorption. In Section 4 we develop an
analysis which explains the numerical results of Section 3. Finally,
some conclusions are shown in Section 5.
2. Atomic model and density-matrix equations
We consider a generic four-level -type atomic system as
shown in Fig. 1 [41,42]. It has a single excited state |2 and three
closely spaced lower levels |1, |3 and |4, the spontaneous emission from excited state |2 to the lower state |1 can strongly affect
the neighbouring transition |2 → |3 (|2 → |4), and interference
due to spontaneous emission can arise. A resonant coupling field
and a weak probe field with Rabi-frequencies Ωc and Ω p , respectively, couple two lower meta-stable states |1 and |3 with
upper level |2, forming a simple -configuration. Meanwhile, the
resulting dark state is coherently coupled to another meta-stable
state |4 by a microwave or quasi-static coherent field with Rabifrequency Ωd . 2γ1 , 2γ3 , 2γ4 denote the spontaneous decay rates
from level |2 to levels |1, |3 and |4, respectively. ω p , ωc and
ωd are carrier frequencies of the corresponding fields, and Δ1 =
ω21 − ω p and Δ3 = ω34 − ωd are the frequency detunings of the
probe field and the coherent driving field, respectively.
Under the rotating-wave approximation, the systematic density
matrix in the interaction picture involving the SGC can be written
as
(1)
The above equations are constrained by ρ11 + ρ22 + ρ33 + ρ44 =
1 and ρ ∗ji = ρi j . In this Letter, in order to distinguish the condition of Ref. [41], we assume the coherent control field is resonant, i.e., Δ3 = 0 and the other parameters are the same. The
√
term 2η γ1 γ3 represents the quantum interference effect from
the cross-coupling between the spontaneous emissions |2 ↔ |1
√
and |2 ↔ |3, and the term 2η γ1 γ4 represents the quantum
interference effect between the spontaneous emission pathways
from |2 ↔ |1 and |2 ↔ |4. The parameter η denotes the align 21 and μ
2 j ) and is defined
ment of the dipole matrix elements (μ
21 · μ
2 j /|μ
21 ||μ
2 j | = cos θ ( j = 3, 4), and θ represent the
as η ≡ μ
21 and μ
2 j . If the maangle between the two dipole moments μ
21 and μ
2 j are orthogonal to each other, i.e., η = 0,
trix elements μ
the SGC disappears. So the existence of the SGC depends on the
21 and μ
2 j . Here we
nonorthogonality of the matrix elements μ
only consider the case where each field acts only on one transition and the polarization direction of the driving field is parallel
to the polarization direction of the coupling field. Then the Rabi
frequencies Ω p , Ωc and Ωd are connected to the angle θ and rep(0)
(0)
1 − η2 (l = p , c , d). To gain the
resented by Ωl = Ωl sin θ = Ωl
remarkable SGC effect on optical properties of the system we assume that three lower levels |1, |3 and |4 are nearly degenerate,
i.e., ω13 ≈ 0 and ω14 ≈ 0 or else the terms with η and accompanied by an exponential factor exp(±ω13 t ) and exp(±ω14 t ), which
are not shown here, have been averaged out to zero and thereby
the SGC effect vanishes [20,30,43].
As is known, the response of the atomic medium to the probe
field is governed by its polarization P = ε0 ( E p χ + E ∗p χ ∗ )/2, with
χ being the susceptibility of the atomic medium. By performing a quantum average of the dipole moment over an ensemble
of N atoms, we find P = N (μ42 ρ24 + μ24 ρ42 ). In order to derive the linear and nonlinear susceptibilities, we need to obtain
the steady-state solution of the density matrix equation. In the
present approach, an iterative method is used and the density ma(0)
(1)
(2)
(3)
trix elements are expressed as ρmn = ρmn + ρmn + ρmn + ρmn + · · · .
Assuming that the coupling field and the driving field are much
stronger than the probe field, the zeroth order solution will be
(0)
ρ11
= 1 and other elements are equal to zero. Under the weakprobe approximation, we can get the matrix element ρ12 up to the
third order:
(1)
ρ21
=
(2)
ρ12
=
(Δ21 − Ωd2 )Ω p
−i (γ1 + γ2 + γ3 )(Δ21 − Ωd2 ) + Δ1 (Δ21 − Ωc2 − Ωd2 )
√
√
(1)
(1)
(η γ1 γ3 Δ1 + η γ1 γ4 Ωd )(ρ12
− ρ21
)Ωc Ω p
−i (γ1 + γ3 + γ4 )(Δ21 − Ωd2 ) + Δ1 (Ωc2 + Ωd2 ) − Δ31
√
√
(2)
(2)
i (η γ1 γ3 Δ1 + η γ1 γ4 Ωd )(ρ12 − ρ21 )Ωc Ω p
(3)
ρ12
=−
−i Δ1 Ωc2 + (γ1 + γ3 + γ4 − i Δ1 )(Ωd2 − Δ21 )
(2)
−
(2)
(2)
,
(2a)
,
(2b)
(2)
i (ρ11 − ρ22 )(Ωd2 − Δ21 )Ω p + i (Δ1 ρ23 + Ωd ρ24 )Ωc Ω p
−i Δ1 Ωc2 + (γ1 + γ3 + γ4 − i Δ1 )(Ωd2 − Δ21 )
.
(2c)
ρ̇11 = 2γ1 ρ22 − i Ω p (ρ12 − ρ21 ),
ρ̇12 = −(γ1 + γ3 + γ4 − i Δ1 )ρ12 − i Ωc ρ13 + i Ω p (ρ22 − ρ11 ),
√
ρ̇13 = i Δ1 ρ13 − i Ωc ρ12 − i Ωd ρ14 + i Ω p ρ23 + 2η γ1 γ3 ρ22 ,
√
ρ̇14 = i Δ1 ρ14 − i Ωd ρ13 + i Ω p ρ24 + 2η γ1 γ4 ρ22 ,
ρ̇22 = −(2γ1 + 2γ3 + 2γ4 )ρ22 + i Ωc (ρ32 − ρ23 ) + i Ω p (ρ12 − ρ21 ),
ρ̇23 = −(γ1 + γ3 + γ4 )ρ23 + i Ωc (ρ33 − ρ22 ) − i Ωd ρ24 + i Ω p ρ13 ,
Therefore, the first-and third-order susceptibility χ (1) and
χ (3)
should be
χ (1) =
χ (3) =
12 |2 (1)
−2N |μ
ρ12 ,
ε0 h̄Ω p
12 |4
−2N |μ
3ε0 h̄3 Ω p3
(3)
ρ12
,
(3)
(4)
χ is be defined as
ρ̇24 = −(γ1 + γ3 + γ4 )ρ24 + i Ωc ρ34 − i Ωd ρ23 + i Ω p ρ13 ,
and
ρ̇33 = 2γ3 ρ22 + i Ωc (ρ23 − ρ32 ) + i Ωd (ρ43 − ρ34 ),
χ = χ (1) + 3| E p |2 χ (3) .
(5)
6458
X.-a. Yan et al. / Physics Letters A 372 (2008) 6456–6460
Fig. 2. The Kerr nonlinearity Re[χ (3) ] (solid curve), linear absorption Im[χ (1) ] (dotted curve), and nonlinear absorption Im[χ (3) ] (dash-dotted curve) of the four-level -type
(0)
system with different SGC. Parameters are Ωc
(0)
= 1.0γ1 , Ωd = 0.2γ1 , γ3 = γ4 = γ1 .
3. Discussion and results
In this part, we will focus on the dependence of the thirdorder susceptibility on the SGC. According to the expressions (3)
and (4), we show in Fig. 2 the linear absorption (dotted curve)
and the refractive part of the third-order susceptibility (solid
curve) as a function of the probe detuning. From Eq. (2a), we
can find the linear absorption is independent of the SGC (i.e.
η), the nonlinear absorption given by the imaginary part of χ (3) ,
which is dependent of η from Eq. (2c), should be considered, so
we also plot the imaginary part of the third-order susceptibility
(dashed curve) in Fig. 2. For simplicity, we scale all the parame(0)
(0)
ters by the decay rate γ1 , setting Ωc = Ωc sin θ = Ωc
1 − η2 ,
(0)
(0)
(0)
1 − η2 and the parameters Ωc = 1.0γ1 ,
Ωd = Ωd sin θ = Ωd
(0)
Ωd = 0.2γ1 , γ3 = γ4 = γ1 . From this figure, we can see that when
η = 0, which means no interference between spontaneous emis-
sion channels, a couple of general linear absorption and Kerr nonlinearity curves occur [39,44]. In other words, if the SGC is absent,
the maximal Kerr nonlinearity is accompanied by strong linear and
nonlinear absorptions, as can be seen from Fig. 2(a), implying that,
in this case, it is not desirable for applications of all-optical switch
because the accompanying thermal effect of the devices is not negligible. In order to eliminate the thermal effect, Ref. [41] proposed
a scheme for giant enhancement of the Kerr nonlinearity by proper
tuning of the coherent driving field in the same four-level -type
system without SGC. Here, when the SGC is taken into account, as
Fig. 3. Variation of Re[ F 1 (η)] versus the probe detuning Δ1 . Parameters are the
same as in Fig. 2.
shown in Figs. 2(b) and (c), a giant Kerr nonlinearity with vanishing absorption also can be obtained under the condition of keeping
the coherent driving field resonant. Moreover, the Kerr nonlinearity is enhanced gradually as the SGC changes from η = 0 to 0.9.
Compared with the case of η = 0, the maximal value of the Kerr
nonlinearity is enhanced by about five times at η = 0.9, which
is different from the scheme proposed by Ref. [41], in which the
X.-a. Yan et al. / Physics Letters A 372 (2008) 6456–6460
Fig. 4. Variation of Re[χ (3) ] (solid curve) and Re[ F 1 (η)] (dotted curve) versus the probe detuning Δ1 . Parameters are the same as in Fig. 2; (a)
maximal value of Re[χ (3) ] is invariable when the coherent driving field is detuned. On the other hand, we find, in the case of
η = 0.9, that the enhanced Kerr nonlinearity is located in the nonlinear gain region and the linear absorption is zero near zero probe
detuning. So in the case, we achieve giant Kerr nonlinearity accompanied by zero linear absorption and negative nonlinear absorption
via SGC.
4. Qualitative explanation of above numerical results
In this part, we will provide a qualitative explanation for the
above numerical results. Here, we mainly discuss the effect of SGC
(2)
on the Kerr nonlinearity. In Eq. (2), since the matrix elements ρ11 ,
(2)
(2)
(2)
ρ22
, ρ23 , ρ24 are independent of η , we do not present the an-
alytical expressions. It can be seen from Eq. (2b) that the density
(2)
matrix element ρ12 now is an extra term introduced by SGC. This
term according makes the third-order susceptibility acquire an additional term associated with SGC. As the analytical expression for
(3)
ρ12
shows, the first term is the product of the coupling field and
the additional term. That is to say, in the present generic four-level
-type atomic system, the spontaneous decays from the common
excited state to the lower closely spaced levels interfere and give
rise to an extra coherence between the lower two levels [36,45,
46]. Thereby the coupling field interacts with this extra coherence
and consequently the Kerr nonlinearity is clearly enhanced. Here,
we designate the first term of Eq. (4) as F 1 (η) and another term
as F 2 , then χ (3) = F 1 (η) + F 2 . We plot Re[ F 1 (η)] as a function of
Δ1 in Fig. 3 with different SGC. Obviously, the stronger the SGC
is, the more remarkable the coupling field interacts with this extra
coherence. Hence, we attribute the enhancement of Kerr nonlinearity to the extra coherence between the lower two levels induced
by SGC.
Another a question arises, that is why the peak value of the
enhanced Kerr nonlinearity enters the electromagnetically induced
transparency and whether the position variation of Re[χ (3) ] is
related to SGC. To answer this question, we plot Re[ F 1 (η)] and
Re[χ (3) ] as a function of Δ1 in Fig. 4 with different SGC. Obviously, one can see that the peak position of the enhanced Re[χ (3) ]
is approximately coincident with that of Re[ F 1 (η)]. Hence, when
the optimal SGC effect is taken into account, a giant Kerr nonlin-
6459
η = 0.8, (b) η = 0.9.
earity accompanied by vanishing absorption can be realized near
zero probe detuning even without the condition of proper detuning of the coherent control field.
5. Conclusion
We have investigated the effect of SGC on the Kerr nonlinearity in a generic four-level -type atomic system. The results show
that the spontaneously generated coherence is capable of enhancing the Kerr nonlinearity even without the condition of Ref. [41].
In the case of optimal SGC, both enhanced Kerr nonlinearity and
negligible linear and nonlinear absorption can be realized simultaneously. Compared with the scheme that offered by Ref. [41],
the enhancement of Kerr nonlinearity can be enhanced by some
times even without the condition of proper detuning of the coherent control field. Close inspection of the analytical expression
undoubtedly shows that the spontaneously emission interference
induces an extra coherence term that is responsible for the large
enhancement of the Kerr nonlinearity, suggesting the application
of the above-mentioned results in all-optical switch, frequency
conversion, polarization phase gates and relevant fields appear to
be promising.
Acknowledgements
This work is supported by the National Natural Science Key
Foundation of China (Grant No. 60678005), the foundation for Key
program of Ministry of Education, China (Grant No. 105156), the
For Ying-Tong Education Foundation for Young Teachers in the Institutions of Higher Education of China (Grant No. 101061) and the
Specialized Research Fund for Doctoral Program of Higher Education of China (Grant No. 2050698017).
References
[1] J. Mompart, R. Corbalan, J. Opt. B: Quantum Semiclass. Opt. 2 (2000) R7.
[2] C.R. Li, Y.C. Li, F.K. Men, C.H. Pao, Y.C. Tsai, J.F. Wang, Appl. Phys. Lett. 86 (2005)
201112.
[3] A.E. Allahverdyan, R.S. Gracia, T.M. Nieuwenhuizen, Phys. Rev. E 71 (2005)
046106.
[4] X.J. Fan, N. Cui, H. Ma, A.Y. Li, H. Li, Eur. Phys. J. D 37 (2006) 129.
[5] H.R. Xia, C.Y. Ye, S.Y. Zhu, Phys. Rev. Lett. 77 (1996) 1032.
6460
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
X.-a. Yan et al. / Physics Letters A 372 (2008) 6456–6460
C.H. Keitel, Phys. Rev. A 57 (1998) 1412.
L.J. Wang, A. Kuzmich, A. Dogariu, Nature 406 (2000) 277.
L.V. Hau, S.E. Harris, Z. Dutton, C.H. Behroozi, Nature (London) 397 (1999) 594.
M.M. Kash, V.A. Sautenkov, A.S. Zibrov, L. Hollberg, G.R. Welch, M.D. Lukin,
Y. Rostovtsev, E.S. Fry, M.O. Scully, Phys. Rev. Lett. 82 (1999) 5229.
M.O. Scully, Phys. Rev. Lett. 67 (1991) 1855.
Q.Y. He, T.J. Wang, J.Y. Gao, Chin. Phys. 15 (2006) 1798.
M. Sahrai, M. Mahmoudi, R. Kheradmand, Phys. Lett. A 367 (2007) 408.
E. Arimondo, in: E. Wolf (Ed.), Progress in Optics XXXV, North-Holland, Amsterdam, 1996, p. 257.
S.E. Harris, Phys. Today 50 (1997) 36.
S.E. Harris, Phys. Rev. Lett. 82 (1999) 4611.
A.G. Litvak, M.D. Tokman, Phys. Rev. Lett. 88 (2002) 095003.
Y. Wu, X.X. Yang, Phys. Rev. A 71 (2005) 053805.
J. Javanainen, Europhys. Lett. 17 (1992) 407.
S. Menon, G.S. Agarwal, Phys. Rev. A 57 (1998) 4014.
E. Paspalakis, S.Q. Gong, P.L. Knight, Opt. Commun. 152 (1998) 293.
S. Menon, G.S. Agarwal, Phys. Rev. A 61 (1999) 013807.
R.M. Whitley, C.R. Stround Jr., Phys. Rev. A 14 (1976) 1498.
Z. Ficek, B.J. Dalton, P.L. Knight, Phys. Rev. A 51 (1995) 4062.
P. Zhou, S. Swain, Phys. Rev. Lett. 78 (1997) 832.
S.Q. Gong, E. Paspalakis, P.L. Knight, J. Mod. Opt. 45 (1998) 2433.
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
[44]
[45]
[46]
F. Ghafoor, S.Y. Zhu, M.S. Zubairy, Phys. Rev. A 62 (2000) 013811.
H.M. Ma, S.Q. Gong, C.P. Liu, Z.R. Sun, Z.Z. Xu, Opt. Commun. 223 (2003) 97.
H.M. Ma, S.Q. Gong, Z.R. Sun, R.X. Li, Z.Z. Xu, Chin. Phys. 15 (2006) 2588.
C.P. Liu, S.Q. Gong, X.J. Fan, Z.Z. Xu, Opt. Commun. 231 (2004) 289.
C.P. Liu, S.Q. Gong, X.J. Fan, Z.Z. Xu, Opt. Commun. 239 (2004) 383.
J.H. Li, J.M. Luo, W.X. Yang, J.C. Peng, Chin. Phys. 14 (2005) 985.
J.H. Wu, J.Y. Gao, Phys. Rev. A 65 (2002) 063807.
P.R. Berman, Phys. Rev. A 72 (2005) 035801.
M.A. Antón, O.G. Calderón, F. Carreño, Phys. Rev. A 72 (2005) 023809.
I. Gonzalo, M.A. Antón, F. Carreño, O.G. Calderón, Phys. Rev. A 72 (2005)
033809.
Y.P. Niu, S.Q. Gong, Phys. Rev. A 73 (2006) 053811.
Y.F. Ban, H. Guo, D.A. Han, H. Sun, J. Opt. B 7 (2005) 35.
R. Arun, Phys. Rev. A 77 (2008) 033820.
S.Q. Gong, Y. Li, S.D. Du, Z.Z. Xu, Phys. Lett. A 59 (1999) 43.
W.H. Xu, J.H. Wu, J.Y. Gao, Phys. Rev. A 66 (2002) 063812.
Y.P. Niu, S.Q. Gong, R.X. Li, Z.Z. Xu, X.Y. Liang, Opt. Lett. 30 (2005) 3371.
M.D. Lukin, S.F. Yelin, M. Fleischhauer, M.O. Scully, Phys. Rev. A 60 (1999) 3225.
B.P. Hou, S.J. Wang, W.L. Yu, W.L. Sun, Phys. Rev. A 69 (2004) 053805.
H. Wang, D.J. Goorskey, M. Xiao, J. Mod. Opt. 49 (2002) 335.
M.O. Scully, Phys. Rev. Lett. 55 (1985) 2802.
J. Evers, D. Bullock, C.H. Keitel, Opt. Commun. 209 (2002) 173.