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Manipulating multi-frequency light in a five-level
cascade-type atomic medium associated with
giant self-Kerr nonlinearity
ANH NGUYEN TUAN,1,2 DOAI LEVAN,1AND BANG NGUYEN HUY1,*
1Vinh University, 182 Le Duan Street, Vinh City, Vietnam
2Ho Chi Minh City University of Food Industry, Ho Chi Minh City, Vietnam
*Corresponding author: bangnh@vinhuni.edu.vn
Received 8 January 2018; revised 18 March 2018; accepted 6 April 2018; posted 11 April 2018 (Doc. ID 319164); published 2 May 2018
We propose a model to manipulate group velocity of a multi-frequency probe light in an electromagnetically
induced transparency medium consisting of five-level cascade-type atoms associated with a giant self-Kerr
nonlinearity. An analytic expression of group index for the probe light is derived as a function of parameters
of the probe and coupling fields, atomic density, and lifetimes of excited atomic states. In the presence of
the self-Kerr, both probe and controlling fields can be used as knobs to manipulate the probe light between
the subluminal and superluminal propagation modes in three separated frequency regions. The theoretical model
agrees with experimental observation, and it is helpful to find the optimized parameters and related applications.
Furthermore, by using such a cascade excitation scheme, it could be possible to choose the uppermost excited
electronic states having long lifetimes, as Rydberg states, to slow the light down to a few mm/s. © 2018 Optical
Society of America
OCIS codes: (020.1670) Coherent optical effects; (190.3270) Kerr effect; (350.5500) Propagation.
https://doi.org/10.1364/JOSAB.35.001233
1. INTRODUCTION
Controlling group velocity of light has been one of the most
interesting topics in optical science during the last two decades
due to its having potential important applications, such as con-
trollable optical delay lines, optical switching, telecommunica-
tion, interferometry, optical data storage and optical memories,
quantum information processing, and so on [1–4]. In general,
slow light or subluminal propagation takes place in a normal
dispersive medium, while fast light or superluminal propaga-
tion is associated with an anomalous dispersive medium.
The advent of electromagnetically induced transparency
(EIT) delivers media of reduced resonant absorption and steep
dispersion for a probe light [5]. Furthermore, magnitude and
sign of dispersion of the medium can be controlled by a
coupling light. Using the EIT technique, several researchers at-
tempted to demonstrate experimentally the subluminal [6–10]
and superluminal [11–14] propagation of light. Other studies
concerned switching between the subluminal and superluminal
light propagation in an atomic medium by changing frequency,
intensity, phase, and polarization of applied fields [15–25].
In addition to steep dispersion, another interesting property
of the EIT medium is exhibition of giant Kerr nonlinearity with
controllable magnitude and sign [26,27]. As a consequence,
such a giant Kerr nonlinearity influences the group velocity
of light in the EIT media, even at very low light intensity.
Indeed, Agarwal et al. [28] demonstrated that cross-Kerr non-
linearity makes a significant contribution to the group velocity.
More recently, Ali et al. [29] showed that the light could be
further slowed by the cross-Kerr nonlinearity. In addition to
the cross-Kerr, a giant self-Kerr nonlinearity also arises in
the EIT medium [5], but it is still not taken into account
in slowing-light studies so far. In addition, a lack of precise de-
scription of the group velocity of light hampers applications
concerning manipulation of light in EIT media.
In the early years of EIT study, most interest focused on the
three-level systems to create a single EIT window in which the
light can be controlled only in a narrow frequency region. From
a practical perspective, extension from a single- to multi-
window EIT is apparently of interest due to its promising ap-
plications in multichannel optical communication, waveguides
for optical signal processing, and multichannel quantum
information processing. A possible way to obtain a multi-
EIT window is to use additionally controlling fields to excite
various multi-level atomic systems [30,31]. Some researchers
used such a technique to control the group velocity at multiple
frequencies [19,23,31,32].
Another simpler way is to use only a controlling field to cou-
ple simultaneously several closely spaced hyperfine levels, which
Research Article Vol. 35, No. 6 / June 2018 / Journal of the Optical Society of America B 1233
0740-3224/18/061233-07 Journal © 2018 Optical Society of America
was first demonstrated in a five-level cascade system [33]. The
first advantage of the five-level cascade scheme is that it is pos-
sible to simultaneously slow probe light at different frequencies
by controlling a sole coupling light. As discussed in Ref. [34],
such slowed light has an advantage in producing the quantum
entanglement. The second advantage is that it is possible to
choose the uppermost excited states as the Rydberg states hav-
ing long lifetimes (see Ref. [23]), which increases significantly
atomic coherence or slower group velocity of the light. The
third advantage arises from the presence of self-Kerr, so one
may control between sub- and superluminal propagation
modes by probe and/or coupling fields. This system has been
studied recently by using an analytic method [35,36] and ex-
tended later to several applications, e.g., enhancement of Kerr
nonlinearity [37,38], optical bistability (OB) [39], generating
optical nano-fibers for guiding entangled beams [40], and op-
tical soliton formation of laser pulses [41]. The analytic model
has been used recently to interpret the experimental observa-
tions with a good agreement [42]. Growing from this interest,
in this work, we propose to use a five-level cascade atomic
medium to control a multi-frequency probe light by a sole light
in the presence of giant self-Kerr nonlinearity. A possible way to
switch between the subluminal and superluminal propagation
mode is discussed.
2. THEORETICAL MODEL
We consider a cold atomic medium consisting of five-level
cascade-type systems, as shown in Fig. 1. A weak probe laser
beam (with frequency ωp) excites the transition j1i↔j2i,
whereas an intense controlling laser beam (with frequency ωc)
couples simultaneously transitions between the state j2iand
three closely spaced states j3i,j4i, and j5i. We denote δ1
and δ2as frequency separations between the levels j3i−j4i
and j5i−j3i, respectively.
The frequency detuning of the probe and controlling lasers
are, respectively, defined as
Δpωp−ω21,Δcωc−ω32 :(1)
In the framework of semiclassical theory, using the dipole
and rotating wave approximations, the evolution of the system
can be represented by the following density-matrix
equations [35]:
_
ρ55 −Γ52ρ55 −
i
2Ωca52ρ25 −ρ52 ,(2)
_
ρ44 −Γ42ρ44 −
i
2Ωca42ρ24 −ρ42 ,(3)
_
ρ33 −Γ32ρ33 −
i
2Ωca32ρ23 −ρ32 ,(4)
_
ρ22 −Γ21ρ22 Γ32 ρ33 Γ42ρ44 Γ52 ρ55
−
i
2Ωpρ12 −ρ21−
i
2Ωca32ρ32 −ρ23
−
i
2Ωca42ρ42 −ρ24 −
i
2Ωca52ρ52 −ρ25 ,(5)
_
ρ11 Γ21ρ22 −
i
2Ωpρ21 −ρ12,(6)
_
ρ54 −iδ1δ2−γ54ρ54 i
2Ωca42ρ52 −
i
2Ωca52ρ24 ,
(7)
_
ρ53 −iδ2−γ53ρ53 i
2Ωca32ρ52 −
i
2Ωca52ρ23 ,(8)
_
ρ52 iΔc−δ2−γ52ρ52 i
2Ωpρ51 i
2Ωca32ρ53
i
2Ωca42ρ54 i
2Ωca52ρ55 −ρ22 ,(9)
_
ρ51 iΔcΔp−δ2−γ51ρ51 i
2Ωpρ52 −
i
2Ωca52ρ21 ,
(10)
_
ρ43 −iδ1−γ43ρ43 −
i
2Ωca42ρ23 i
2Ωca32ρ42 ,(11)
_
ρ42 iΔcδ1−γ42ρ42 i
2Ωpρ41 i
2Ωca32ρ43
i
2Ωca52ρ45 i
2Ωca42ρ44 −ρ22 ,(12)
_
ρ41 iΔcΔpδ1−γ41ρ41 i
2Ωpρ42 −
i
2Ωca42ρ21 ,
(13)
Fig. 1. Five-level cascade excitation scheme.
1234 Vol. 35, No. 6 / June 2018 / Journal of the Optical Society of America B Research Article
_
ρ32 iΔc−γ32ρ32 i
2Ωpρ31 i
2Ωca32ρ33 −ρ22
i
2Ωca42ρ34 i
2Ωca52ρ35 ,(14)
_
ρ31 iΔcΔp−γ31ρ31 i
2Ωpρ32 −
i
2Ωca32ρ21 ,(15)
_
ρ21 iΔp−γ21ρ21 i
2Ωpρ22 −ρ11−
i
2Ωca32ρ31 ,
−
i
2Ωca42ρ41 −
i
2Ωca52ρ51 ,(16)
ρki ρ
ik,(17)
ρ11 ρ22 ρ33 ρ44 ρ55 1,(18)
where iis the complex number; Ωpd21Ep∕ℏand Ωc
d32Ec∕ℏare Rabi frequencies; dkl is an element of dipole
moment of the jki−jlitransition; a32 d32∕d32 ,
a42 d42∕d32 , and a52 d52 ∕d32 are the relative transition
strengths; and γkl is the decay rate of the atomic coherence ρkl ,
given by [35]
γkl 1
2X
Ek<Ej
Γjk X
Em<El
Γlm,(19)
where Γkl is the decay rate of population from level jkito
level jli.
In a steady regime, the solution for the matrix element ρ21
can be calculated up to the third-order as [37]
ρ21 ρ1
21 ρ3
21 −
iΩp
2FiΩp
2FΩ2
p
2Γ21 1
F1
F,(20)
where
Fγ21 −iΔpa2
32Ωc∕22
γ31 −iΔpΔc
a2
42Ωc∕22
γ41 −iΔpΔcδ1,a2
52Ωc∕22
γ51 −iΔpΔc−δ2,
(21)
and Fis conjugation of F.
The total susceptibility can then be determined by the fol-
lowing relation:
χ−2Nd21
ε0Ep
ρ21,(22)
where Nis the density of particles and ε0is the permittivity in
vacuum.
In order to extract the first- and third-order susceptibilities,
we interpret the total susceptibility in Eq. (22) in an alternative
form:
χχ13E2
pχ3:(23)
Finally, the first-order and third-order susceptibilities are
given as [37]
χ1Nd2
21
ε0ℏA
A2B2iB
A2B2,(24)
χ3−
Nd4
21
3ε0ℏ3
1
Γ21
B
A2B2A
A2B2iB
A2B2,(25)
where Aand Bare controllable parameters given by
A−ΔpA32
γ31
A42
γ41
A52
γ51
,(26)
Bγ21 A32
ΔpΔc
A42
ΔpΔcδ1
A52
ΔpΔc−δ2
,
(27)
A32 γ31ΔpΔc
γ2
31 ΔpΔc2a2
32Ωc∕22,(28)
A42 γ41ΔpΔcδ1
γ2
41 ΔpΔcδ12a2
42Ωc∕22,(29)
A52 γ51ΔpΔc−δ2
γ2
51 ΔpΔc−δ22a2
52Ωc∕22:(30)
From the linear and third-order susceptibilities, the linear
index n0and self-Kerr nonlinear coefficient n2for the probe
light are derived as
n01Reχ1
21Nd2
21
2ε0ℏ
A
A2B2,(31)
n23Reχ3
4ε0n2
0c
−
Nd4
21
4ε2
0ℏ3c
1
Γ21
AB
1Nd2
21
2ε0ℏ
A
A2B2A2B22
:(32)
In the case of absence of self-Kerr nonlinearity, the group
index is determined by
n0
gn0ωp
∂n0
∂ωp
≃ωp
Nd2
21
2ε0ℏA0A2B2−2AAA0BB 0
A2B22,(33)
where A0and B0represent the derivatives of Aand Bover ωp,
respectively:
A0−1A32
γ31ΔpΔc
−
2A2
32
a2
32Ωc∕22γ2
31
A42
γ41ΔpΔcδ1
−
2A2
42
a2
42Ωc∕22γ2
41
A52
γ51ΔpΔc−δ2
−
2A2
52
a2
52Ωc∕22γ2
51
,(34)
Research Article Vol. 35, No. 6 / June 2018 / Journal of the Optical Society of America B 1235
B0−
2A2
32
a2
32Ωc∕22γ31 ΔpΔc
−
2A2
42
a2
42Ωc∕22γ41 ΔpΔcδ1
−
2A2
52
a2
52Ωc∕22γ51 ΔpΔc−δ2:(35)
It is noted that there are three EIT windows centered at
probe frequencies given by the two-photon resonance condi-
tions: ΔpΔc0,ΔpΔcδ10, and ΔpΔc−δ20
[35]. Therefore, the group index in each EIT window can be
approximated by
n0
gj32 ≃ωp
∂n0
∂ωp
ΔpΔc0
2ωpNd2
21
ε0ℏ
a2
32Ω2
c−4γ2
31
a2
32Ω2
c4γ21γ31 2,
(36)
n0
gj42 ≃ωp
∂n0
∂ωp
ΔpΔcδ10
2ωpNd2
21
ε0ℏ
a2
42Ω2
c−4γ2
41a2
42Ω2
c4γ21γ41 2
−4δ1γ412
4δ1γ412a2
42Ω2
c4γ21γ41 22,
(37)
n0
gj52 ≃ωp
∂n0
∂ωp
ΔpΔc−δ20
,
2ωpNd2
21
ε0ℏ
a2
52Ω2
c−4γ2
51a2
52Ω2
c4γ21γ51 2
−4δ2γ512
4δ2γ512a2
52Ω2
c4γ21γ51 22,
(38)
where n0
gj32,n0
gj42, and n0
gj52 are the group index at the EIT
window in which the controlling field induces the transitions
j2i↔j3i,j2i↔j4i, and j2i↔j5i, respectively.
In the presence of the self-Kerr nonlinearity, the effective
index for the probe light (with intensity Ip) is determined by
nn0n2Ip:(39)
The group index under presence of self-Kerr nonlinearity
can therefore be determined as [43]
nK
gnωp
∂n
∂ωp
n0n2Ipωp∂n0
∂ωp
∂n2
∂ωp
Ip,
(40)
where n0and n2are the linear and nonlinear refractive indices
determined by Eqs. (31) and (32), respectively.
3. APPLICATION TO 85Rb ATOMIC MEDIUM
In order to illustrate applications of the analytic model,
we apply to a cold atomic medium of 85 Rb, where the
Doppler effect can be ignored. The states, j1i,j2i,j3i,j4i,
and j5i, are chosen as 5S1∕2F3,5P3∕2F03,
5D5∕2F00 3,5D5∕2F00 4, and 5D5∕2F00 2, re-
spectively. The atomic parameters are given [35,44]: N
5×1011 atoms∕cm3;Γ32 Γ42 Γ52 2π×0.97 MHz;
Γ21 2π×6 MHz;δ12π×9 MHz;δ22π×7.6 MHz;
d21 1.6×10−29 C·m;ωp2π×3.77 ×108MHz; and
a32:a42 :a52 1:1.4:0.6.
In order to see variations of linear and nonlinear indices, we
plotted the linear index n0and Kerr nonlinearity n2versus the
probe frequency detuning Δpfor the fixed parameters Ωc
10 MHz and Δc0, as in Fig. 2. From Fig. 2, we can see
a normal and anomalous dispersion of the linear index (dashed
curve) in three separated regions corresponding to three EIT
windows that center at Δp−9MHz,Δp0, and Δp
7.6 MHz [35]. Such a dispersive property delivers both sub-
and superluminal propagation modes for the multi-frequency
probe light. On the other hand, the dispersion of the linear
index is in the opposite direction from that of the self-Kerr non-
linearity (solid curve in Fig. 2); thus, the giant self-Kerr non-
linearity can reduce the effective index or enhance the group
velocity of the probe light [see Fig. 3(a)]. Furthermore, the
self-Kerr nonlinearity leads to variation of both magnitude
and sign of the group index under changing probe intensity;
consequently, one may manipulate the probe light to propagate
between sub- and superluminal mode by tuning its own inten-
sity [Fig. 3(b)]. This variation can be explained from Eq. (40)
in that the group index depends proportionally on intensity of
the probe light; thus, the self-Kerr nonlinearity is more efficient
with high intensity of the probe light.
In Fig. 4, we consider influence of the coupling field on the
group index by plotting the nK
gversus the frequency detuning
Δc(a) and the Rabi frequency Ωc(b) for three EIT windows
centered at Δp−2,Δp−10 MHz, and Δp8 MHz.Itis
shown that the group index varies between negative and pos-
itive values with changing intensity and/or frequency of the
coupling field. In other words, one may manipulate the probe
light to propagate between sub- and superluminal mode by
controlling and/or frequency of the coupling field. This can
be explained by noticing that magnitude and sign of both linear
and nonlinear indices depend sensitively on intensity and/or
Fig. 2. Variations of the self-Kerr nonlinearity n2(solid) and linear
index of refraction n0(dashed) versus the probe frequency detuning Δp
when Ωc10 MHz and Δc0.
1236 Vol. 35, No. 6 / June 2018 / Journal of the Optical Society of America B Research Article
frequency of the coupling field [35,37]. On the other hand, it
should be noted from Fig. 4(a) that variation of coupling fre-
quency while keeping probe frequency may cause significant
absorption outside the EIT widows [35].
4. DISCUSSION
In principle, the five-level cascade-type scheme can be used for
any atomic or molecular system having the energy structure
as in Fig. 1. At first, we showed an advantage of the analytic
model for estimating an achievable minimum group velocity
of the probe light. Indeed, based on Eq. (36), the group index
(or the group velocity) is maximized (or minimized) at the fol-
lowing Rabi frequency:
Ωc2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
γ21 2γ31γ31
p:(41)
At such Rabi frequency, the minimum group velocity in the
first EIT window is determined as
v0
g32min 8ε0ℏc
ωpNd2
21
γ21 γ31γ31 :(42)
It is shown from Eq. (42) that the group velocity depends on
the damping rate γ31 or depends inversely on lifetimes of the
excited electronic states. For the excited states having lifetimes
of a few ns, the minimum group velocity can be a few m/s, as
shown in Ref. [6]. However, the cascade excitation scheme
delivers a possible way to choose the uppermost excited states
as the Rydberg states, which have lifetimes of a few μs (lifetime
Fig. 3. (a) Variation of the group index versus probe frequency de-
tuning in the case of self-Kerr nonlinearity absent (dashed) and present
(solid) when Ip10 mW∕cm2,Ωc4 MHz, and Δc0; the dot-
ted curve represents EIT spectrum plotted from the imaginary part of
Eq. (24). (b) Variation of the Kerr nonlinearity nK
gversus probe in-
tensity Ipwhen Ωc4 MHz and ΔcΔp0.
Fig. 4. Variations of nK
gversus Δcwhen Ωc4 MHz (a) and
versus Ωcwhen Δc0(b) at Ip10 mW∕cm2and Δp−2 MHz
(solid), Δp−10 MHz (dashed), and Δp8 MHz (dotted).
Research Article Vol. 35, No. 6 / June 2018 / Journal of the Optical Society of America B 1237
of the state 38D5∕2of Rb atom is 13 μs, see Ref. [45], e.g.). In
this case, one may slow down the probe light to a few mm/s
(ultraslow light). On the other hand, from Eq. (42), one may
further slow down group velocity of the probe light by increas-
ing the atomic density.
From a practical aspect, it is worth to evaluate the value of
transparency efficiency at which the group index maximizes.
For the EIT window at Δp0, the transparency efficiency
is given as [35]
R32 α0−αΩc
α0a2
32Ω2
c
4γ21γ31 a2
32Ω2
c
,(43)
where α0and αΩcare the absorption coefficient when the
controlling laser turns off and on. Substituting Eq. (41)into
Eq. (43), we have
R32 1
21γ31
γ21 γ31:(44)
On the other hand, from Eqs. (36) and (43), we found an
expression of the group index as a function of the transparency
efficiency as follows:
n0
g32 2ωpNd2
21
ε0ℏR321−R32
4γ21γ31
−
1−R322
4γ2
21 :(45)
Variation of the group index versus the transparency depth
is plotted in Fig. 5. One can clearly see that the maximum
group index occurs at the transparency efficiency equal to ap-
proximately 60% (for 85Rb atoms), which agrees with Eq. (44).
Finally, we compared the theoretical result with a prominent
observation in Ref. [6] by restricting the coupling parameters
A52 A42 0in Eqs. (33) and (40) to reduce the five- to
three-level excitation scheme. The group velocity is plotted ver-
sus the coupling Rabi frequency under the presence (solid) and
absence (dashed) of the self-Kerr nonlinearity, where all param-
eters are given as the same as in Ref. [6], as shown in Fig. 6.It
should be noted that the measured value of group velocity
vg17 m∕sis attained at Ic12 mW∕cm2(corresponding
Ωc5.3 MHz), whereas the theoretical value at the same
parameters is vg17 m∕sor vg15 m∕sfor the presence
or absence of the self-Kerr nonlinearity, respectively. This
comparison shows that the model is more accurate with the
inclusion of the self-Kerr nonlinearity. Indeed, whenever the
self-Kerr nonlinearity is excluded, deviation will be greater at
higher probe intensity, which is 10% at Ip5mW∕cm2.On
the other hand, Fig. 6shows a possible optimization to further
slow down the probe light by reducing its own intensity to an
ideal case (without influence of the self-Kerr nonlinearity).
5. CONCLUSION
We have proposed a model for manipulation of a multi-
frequency probe light in a five-level cascade-type medium in
the presence of the self-Kerr nonlinearity. The group index
for the probe light is derived as an analytic function of the
parameters of the light fields, atomic density, and atomic elec-
tronic lifetimes. Although the self-Kerr nonlinearity enhances
group velocity, one may use the probe and/or coupling fields as
knobs to manipulate the probe light between the subluminal
and superluminal modes in three separated frequency regions.
The model agrees with experimental observation, and it is help-
ful in finding the optimized parameters and related applica-
tions. Based on the cascade excitation scheme, it could be
possible to choose the uppermost excited electronic states hav-
ing long lifetimes, as Rydberg states, to manipulate group veloc-
ity of light to a few mm/s.
Funding. Vietnam Ministry of Science and Technology
(ĐTĐLCN.17/17).
Fig. 5. Variation of group index n0
g32 versus the transparency effi-
ciency R32 at ΔpΔc0.
Fig. 6. Plot of the group velocity versus the Rabi frequency Ωcof
the coupling field under the presence (solid) and absence (dashed) of
the self-Kerr nonlinearity when Ip5mW∕cm2, and ΔpΔc0.
1238 Vol. 35, No. 6 / June 2018 / Journal of the Optical Society of America B Research Article
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Research Article Vol. 35, No. 6 / June 2018 / Journal of the Optical Society of America B 1239