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Manipulating multi-frequency light in a five-level cascade EIT medium under Doppler broadening

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  • Ho Chi Minh City University of Industry and Trade

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An electromagnetically induced transparency (EIT) medium consisting of five-level-cascade atoms is proposed to control the group velocity of multi-frequency light under Doppler broadening. An analytic expression for the group index of probe light is derived as a function of the parameters of coupling light and temperature of the medium. It is shown that by adjusting intensity and/or frequency of the solely coupling light one may manipulate simultaneously group velocity of the probe light in three separated frequency regions, each of which enables to switch between the subluminal and superluminal modes. On the other hand, the effect of Doppler broadening increases the group velocity and enlarges the tuning range of the coupling intensity to switch between the superluminal and subluminal modes. The model is helpful to find applications related to the group velocity manipulation of multi-frequency light, and to serve future experimental studies at various temperature conditions.
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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 [14]. 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 [610]
and superluminal [1114] 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 [1525].
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 j1ij2i,
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 j3ij4i
and j5ij3i, respectively.
The frequency detuning of the probe and controlling lasers
are, respectively, defined as
Δpωpω21,Δcωcω32 :(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 iδ1δ2γ54ρ54 i
2Ωca42ρ52
i
2Ωca52ρ24 ,
(7)
_
ρ53 iδ2γ53ρ53 i
2Ωca32ρ52
i
2Ωca52ρ23 ,(8)
_
ρ52 iΔcδ2γ52ρ52 i
2Ωpρ51 i
2Ωca32ρ53
i
2Ωca42ρ54 i
2Ωca52ρ55 ρ22 ,(9)
_
ρ51 iΔcΔpδ2γ51ρ51 i
2Ωpρ52
i
2Ωca52ρ21 ,
(10)
_
ρ43 iδ1γ43ρ43
i
2Ωca42ρ23 i
2Ωca32ρ42 ,(11)
_
ρ42 iΔcδ1γ42ρ42 i
2Ωpρ41 i
2Ωca32ρ43
i
2Ωca52ρ45 i
2Ωca42ρ44 ρ22 ,(12)
_
ρ41 iΔcΔpδ1γ41ρ41 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 iΔcγ32ρ32 i
2Ωpρ31 i
2Ωca32ρ33 ρ22
i
2Ωca42ρ34 i
2Ωca52ρ35 ,(14)
_
ρ31 iΔcΔpγ31ρ31 i
2Ωpρ32
i
2Ωca32ρ21 ,(15)
_
ρ21 iΔpγ21ρ21 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; Ωpd21Epand Ωc
d32Ecare Rabi frequencies; dkl is an element of dipole
moment of the jkijlitransition; a32 d32d32 ,
a42 d42d32 , 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
32Ωc22
γ31 iΔpΔc
a2
42Ωc22
γ41 iΔpΔcδ1,a2
52Ωc22
γ51 iΔpΔcδ2,
(21)
and Fis conjugation of F.
The total susceptibility can then be determined by the fol-
lowing relation:
χ2Nd21
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
ε0A
A2B2iB
A2B2,(24)
χ3
Nd4
21
3ε03
1
Γ21
B
A2B2A
A2B2iB
A2B2,(25)
where Aand Bare controllable parameters given by
AΔpA32
γ31
A42
γ41
A52
γ51
,(26)
Bγ21 A32
ΔpΔc
A42
ΔpΔcδ1
A52
ΔpΔcδ2
,
(27)
A32 γ31ΔpΔc
γ2
31 ΔpΔc2a2
32Ωc22,(28)
A42 γ41ΔpΔcδ1
γ2
41 ΔpΔcδ12a2
42Ωc22,(29)
A52 γ51ΔpΔcδ2
γ2
51 ΔpΔcδ22a2
52Ωc22:(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
2ε0
A
A2B2,(31)
n23Reχ3
4ε0n2
0c
Nd4
21
4ε2
03c
1
Γ21
AB
1Nd2
21
2ε0
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
2ε0A0A2B22AAA0BB 0
A2B22,(33)
where A0and B0represent the derivatives of Aand Bover ωp,
respectively:
A01A32
γ31ΔpΔc
2A2
32
a2
32Ωc22γ2
31
A42
γ41ΔpΔcδ1
2A2
42
a2
42Ωc22γ2
41
A52
γ51ΔpΔcδ2
2A2
52
a2
52Ωc22γ2
51
,(34)
Research Article Vol. 35, No. 6 / June 2018 / Journal of the Optical Society of America B 1235
B0
2A2
32
a2
32Ωc22γ31 ΔpΔc
2A2
42
a2
42Ωc22γ41 ΔpΔcδ1
2A2
52
a2
52Ωc22γ51 ΔpΔcδ2:(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 ΔpΔcδ20
[35]. Therefore, the group index in each EIT window can be
approximated by
n0
gj32 ωp
n0
ωp
ΔpΔc0
2ωpNd2
21
ε0
a2
32Ω2
c4γ2
31
a2
32Ω2
c4γ21γ31 2,
(36)
n0
gj42 ωp
n0
ωp
ΔpΔcδ10
2ωpNd2
21
ε0
a2
42Ω2
c4γ2
41a2
42Ω2
c4γ21γ41 2
4δ1γ412
4δ1γ412a2
42Ω2
c4γ21γ41 22,
(37)
n0
gj52 ωp
n0
ωp
ΔpΔcδ20
,
2ωpNd2
21
ε0
a2
52Ω2
c4γ2
51a2
52Ω2
c4γ21γ51 2
4δ2γ512
4δ2γ512a2
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
j2ij3i,j2ij4i, and j2ij5i, 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
n0n2Ipωpn0
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 5S12F3,5P32F03,
5D52F00 3,5D52F00 4, and 5D52F00 2, re-
spectively. The atomic parameters are given [35,44]: N
5×1011 atomscm3;Γ32 Γ42 Γ52 2π×0.97 MHz;
Γ21 2π×6 MHz;δ12π×9 MHz;δ22π×7.6 MHz;
d21 1.6×1029 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 Δp9MHz,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 Δp2,Δp10 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:
Ωc2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
γ21 2γ31γ31
p:(41)
At such Rabi frequency, the minimum group velocity in the
first EIT window is determined as
v0
g32min 8ε0c
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 mWcm2,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 mWcm2and Δp2 MHz
(solid), Δp10 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 38D52of 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 α0αΩc
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
ε0R321R32
4γ21γ31
1R322
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 msis attained at Ic12 mWcm2(corresponding
Ωc5.3 MHz), whereas the theoretical value at the same
parameters is vg17 msor vg15 msfor 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 Ip5mWcm2.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 Ip5mWcm2, 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
... However, most of these works are applied to ultracold atomic media with the Doppler effect being ignored. A small number of other studies were performed at room temperature with the Doppler effect included [19,20,[31][32][33][34][35][36], which is needed for experimental and applied studies in practice. ...
... In addition, we also note that the external magnetic field has no effect on the m F = 0 states, so the transition of the π-polarized light is also not affected by the magnetic field. The atomic parameters can be chosen as [34]: N = 7 × 10 17 atoms m −3 , γ 31 = γ 32 = 5.3 MHz, d 31 = 1.6 × 10 −29 C·m. We also note that the boiling point of rubidium is about 39 • C [37], so that at room temperature it exists in a gaseous state and is therefore influenced by Doppler broadening significantly. ...
... It is shown that, at the probe frequency corresponding to the frequency detuning ∆ p = 3γ, the peak of the group index is reduced as the coupling field intensity increases, at the same time the profile of the group index is expanded due to the power broadening. The increase in laser coupling strength leads to the extension of the EIT window thereby reducing the slope of the dispersion curve [5,34], which is the reason for the decrease in the amplitude of the group refractive index as we see in figure 7(b). Finally, in figure 8 we investigate the influence of Doppler effect (or temperature) on the group index. ...
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The optical response of a magnetic-degenerated four-level atom system to the two left and right circular polarization components of the probe field is represented at room temperature. The absorption spectrum and group index for the two polarization components of the probe field are controlled according to the static magnetic field and the coupling field under electromagnetically induced transparency condition. By varying the strength of the static magnetic field, the optical response of the atomic medium can be changed from transparency to absorption or vice versa and hence the amplitude of group index also changes from positive extreme to negative extreme or vice versa. The same phenomenon also occurs when changing the coupling field intensity. In addition, temperature also significantly influence on the optical response of the atomic medium, which changes not only the amplitude but also the sign of the group index as the temperature increases. Our analytical results can be useful for experimental observation and related applications of light group index/velocity at room temperature.
... Electromagnetic induced transparency (EIT) [7,8] is a very important phenomenon that is based on quantum interference and quantum coherence and leads to controlling the group velocity and controlling the absorption and dispersion properties of light in mediums. So far, many quantum systems have been proposed based on atomic vapor and solid states to study the quantum phenomena in vacuum [9][10][11][12][13][14], but for use in optical devices, it needs to be studied in a non-vacuum environment. The photonic crystal is an important environment in which a quantum system can be studied that will have very important results in improving quantum properties and quantum phenomena [15,16]. ...
... Substituting equation (7) into equations (9)-(10), we obtain the following set of density matrix equations of motion where r r = ij ij * and r r r + + = 1. 11 22 33 Here j j j D = p c is the phase difference between applied field and the terms w w D = p p 13 and w w D = c c 23 are the probe and coupling fields detuning, respectively. In addition, g 31 and g 32 are the spontaneous decay rates from upper level to the lower levels. ...
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We investigate the steady-state and dynamic behavior of the optical properties of the laser pulse in a GaAs/AlAs 1D photonic crystal (1DPC) with an atomic vapor defect layer. We chose the atomic vapor of the ⁸⁷ Rb as a defect layer by driving the probe field on the D 2 transition between 5 ² P 3/2 - 5 ² S 1/2 levels through the 1DPC. The effect of the photonic bandgap (PBG) on the absorption and dispersion properties of the probe field is discussed. Moreover, the transmission and reflection coefficient of the 1DPC is controlled by applying the various value of the intensity of the coupling field. By comparing these results in the vacuum and near the PBG, we find that the absorption/dispersion and transmission/reflection properties are strongly affected by the PBG. We find that all-optical properties of atomic vapor in the surrounding of 1DPC are improved due to PBG. Furthermore, the effect of the intensity of the coupling field on the all-optical switching is studded. The proposed model may provide some new possibilities for technological applications as an all-optical device based on the photonic crystal in quantum information science, quantum computing, signal processing, and quantum communications.
... The advent of EIT [2][3][4][5][6][7] has created transparent media whose optical properties can be controlled by the external fields. Due to very steep dispersion in transparent spectral region, therefore light pulses can propagate with very small group velocities [8][9][10], the medium can achieve giant Kerr nonlinearities [11][12][13][14], and optical soliton is also easily achieved with low intensity [15][16][17][18][19]. ...
... On the other hand, oscillations at the front edge of the probe pulse can be extinguished when increasing width of the signal pulse (Fig. 5b). Here, the frequency detuning of the signal field in Eq. (9) or Eq. (10) is switched with approximate period 50/21 and 100/21, respectively. ...
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We proposed a model for all-optical switching in a medium consisting of four-level vee-cascade atomic systems excited by coupling, probe, and signal fields. It is shown that, by changing the intensity or the frequency of the signal field, the medium can be actively switched between either electromagnetically induced transparency (EIT) or electromagnetically induced absorption (EIA), which has behavior of all-optical switching. As a result, a cw probe field is switched into square pulses by modulating the intensity or the frequency of the signal light. Furthermore, width of the square probe pulses can be controlled by tuning the switching period of the signal field. Such a tuneable all-optical switching is useful for finding related applications in optic communications and optical storage devices.
... Consequently, multilevel atomic systems become of interest in generating multielectromagnetically induced transparency [10][11][12][13][14][15][16][17][18][19][20]. Based on multi-window EIT, one has also created multi-frequency slow or fast light [11,[21][22][23][24]. Experimentally, multi-window EIT has also been observed in some works [12][13][14][15]17,18,20], whereas observations of the group index are relatively modest [25][26][27]. ...
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We report an experimental measurement of giant group index in the ⁸⁵Rb atomic medium at room temperature via a four-level V-type scheme on the D2 line. Our experiment uses co-propagation configuration of probe and coupling lasers through an atomic sample. In this configuration, three sharp EIT windows with significant transparency depths are observed on the probe absorption spectrum. By establishing a Mach–Zehnder interferometer, we measure the dispersive curve and hence obtain the group index curve with three enhanced positive peaks at the locations of the EIT windows, interspersed with negative peaks. The amplitude of the group index curve is increased as the temperature decreases and is decreased as the temperature increases. We estimate from our experimental results the good values of the group index to be ng = 5.8 × 10⁴ (slow light regime) and ng = −4.0 × 10⁴ (fast light regime). We also show that the experimental measurements are in good agreement with the theoretical results.
... Because of a smaller sensitivity to field gradient, one could achieve better sensitivities in high pulsed field measurements [19] by using nanocells. To conclude, we note that our findings could also be useful to enlarge the tuning range of group velocity manipulation for slow-light experiments using electromagnetically-induced transparency [36]. ...
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The Zeeman effect is an important topic in atomic spectroscopy. The induced change in transition frequencies and amplitudes finds applications in the Earth-field-range magnetometry. At intermediate magnetic field amplitude B ∼ B 0 = A hfs /µ B , where A hfs is the magnetic dipole constant of the ground state, and µ B is the Bohr magneton (B 0 ≈ 1.7 kG for Cs), the rigorous rule ∆F = 0, ±1 is affected by the coupling between magnetic sub-levels induced by the field. Transitions satisfying ∆F = ±2, referred to as magnetically-induced transitions, can be observed. Here, we show that a significant redistribution of the Cs 6S 1/2 → 6P 3/2 magnetically-induced transition intensities occurs with increasing magnetic field. We observe that the strongest transition in the group F g = 3 → F e = 5 (σ + polarization) for B < B 0 cease to be the strongest for B > 3 B 0. On the other hand, the strongest transition in the group F g = 2 → F e = 4 (σ − polarization) remains so for all our measurements with magnetic fields up to 9 kG. These results are in agreement with a theoretical model. The model predicts that similar observations can be made for all alkali metals, including Na, K and Rb atoms. Our findings are important for magnetometers utilizing the Zeeman effect above Earth field, following the rapid development of micro-machined vapor-cell-based sensors.
... The advent of EIT (Imamoǧlu and Harris 1989;Boller et al. 1991;Fleischhauer et al. 2005;Doai et al. 2014;Khoa et al. 2016Khoa et al. , 2017a has created transparent media whose optical properties can be controlled by the external fields. Due to very steep dispersion in transparent spectral region, therefore light pulses can propagate with very small group velocities (Hau et al. 1999;Anh et al. 2018aAnh et al. , 2018b, the medium can achieve giant Kerr nonlinearities (Wang et al. 2001;Khoa et al. 2014;Hamedi et al. 2016;Doai et al. 2019), and optical soliton is also easily achieved with low intensity (Huang et al. 2006;Si et al. 2010;Chen et al. 2014;Khoa et al. 2017b;Dong et al. 2018). ω s ) optical fields with Rabi frequency Ω c and Ω s , respectively. ...
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We proposed a model for all-optical switching in a medium consisting of four-level vee-cascade atomic systems excited by coupling, probe, and signal fields. It is shown that, by changing the intensity or the frequency of the signal field, the medium can be actively switched between either electromagnetically induced transparency or electromagnetically induced absorption, which has behavior of all-optical switching. As a result, a cw probe field is switched into square pulses by modulating the intensity or the frequency of the signal light. Furthermore, width of the square probe pulses can be controlled by tuning the switching period of the signal field. Such a tuneable all-optical switching is useful for finding related applications in optic communications and optical storage devices.
... The above studies on negative refraction in EIT materials have often ignored the Doppler effect and thus they can only be suitable for ultra-cooled atoms confined in a magneto-optical trap (MOT), for example. In order to be able to apply EIT materials in practice operating under different temperature conditions, some studies on EIT [31][32][33][34][35][36][37] as well as its applications such as group velocity [38][39][40][41], pulse propagation [42,43], Kerr nonlinearity [44][45][46][47] and so on, have been included the Doppler broadening. It showed that the influence of Doppler broadening is very significant on the EIT effect at room temperature. ...
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We have achieved a negative refractive index with significantly reduced absorption in a three-level Λ-type atomic gas medium under Doppler broadening. It shows that the conditions for obtaining negative refractive index in the presence of Doppler broadening are very different from those of Doppler broadening absent. In particular, in order to obtain negative refractive index in the case of Doppler broadening the coupling laser intensity must be approximately ten times greater than that when the Doppler broadening is ignored. Meanwhile, the frequency band of negative refractive index with Doppler broadening is significantly expanded (about a hundred times) compared to that without Doppler broadening, however, the amplitude of negative refractive index decreases with increasing temperature (or Doppler width). Even in some cases as temperature (Doppler width) increases, the left-handedness of the material can disappear. In addition, we also show that the amplitude and the frequency band of negative refractive index can be changed by adjusting the intensity and the frequency of coupling laser. Our theoretical investigation can be useful for selection of laser parameters under different temperature conditions to achieve negative refractive index in experimental implementation.
... However, the above studies on negative refractive index in EIT materials have often ignored Doppler broadening and thus they can only be suitable for ultra-cooled atoms confined in a magneto-optical trap (MOT). In order to be able to apply EIT materials in practice operating under different temperature conditions, Doppler effect has been included in some recent studies on EIT [16]- [23], group velocity of light [24]- [27], pulse propagation [28], [29], Kerr nonlinearity [30]- [34]. It showed that the influence of Doppler broadening on the EIT Manuscript ID: PJ-012191-2021 and related phenomena at room temperature should be considered. ...
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In this paper, we present the study on negative refractive index in an inhomogeneously broadened four-level inverted-Y atomic medium based on electromagnetically induced transparency (EIT). The expressions of the relative permittivity and the relative permeability are derived under Doppler broadening. For this four-level system, we have found two frequency bands of negative refractive index in an optical region corresponding to two EIT windows. The influences of coupling and signal laser fields as well as temperature on frequency bands of negative refractive index are investigated. Our research can be convenient for experimental implementation with real atomic media under different temperatures.
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We have measured the dispersive profile of a multiwindow electromagnetically induced transparency (EIT) spectrum in a medium consisting of Rb atoms in the presence of Doppler broadening. The atomic medium is excited by a strong coupling light and a weak probe laser light via the V-type transitions within the D 2 manifold. Under the EIT effect, an anomalous dispersive region of the medium is basically modified into multinormal and anomalous dispersive regions. Furthermore, the slope and position of the dispersion can be controlled with the intensity and frequency of the coupling light. An analytic model is proposed to simulate the observed spectrum with a good agreement. Such controllable dispersive properties with their analytic description would be useful for finding applications related to multiwindow EIT phenomena.
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We report the experimental manipulation of the group velocities of reflected and transmitted light pulses in a degenerate two-level atomic system driven by a standing wave, which is created by two counter-propagating light beams of equal frequencies but variable amplitudes. It is shown that the light pulse is reflected with superluminal group velocity while the transmitted pulse propagates from subluminal to superluminal velocities via changing the power of the backward coupling field. We find that the simultaneous superluminal light reflection and transmission can be reached when the power of the backward field becomes closer or equal to the forward power, in this case the periodical absorption modulation for photonic structure is established in atoms. The theoretical discussion shows that the anomalous dispersion associated with a resonant absorption dip within the gain peak due to four-wave mixing leads to the superluminal reflection, while the varying dispersion from normal to anomalous at transparency, transparency within absorption, and electromagnetically induced absorption windows leads to the subluminal to superluminal transmission.
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We propose an optical bistability (OB) model consisting of a five-level cascade electromagnetically induced transparency (EIT) medium placed in a unidirectional ring cavity. An input-output intensity relationship for a weak probe light field is derived as a function of parameters of a strong coupling light and cooperation C. The OB behaviors of the system are investigated with variation of the parameters. It has been shown that the OB behaviors occur simultaneously in three EIT windows. Furthermore, switching thresholds and the hysteresis loop can be controlled by the parameters of the coupling light and/or a cooperation of atomic medium. Such controllable OB behaviors can find interesting applications in bistable photonic devices working at low light intensity and multiple frequencies.
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We develop an analytical approach on electromagnetically induced transparency (EIT) in a Doppler broadened medium consisting of five-level cascade systems excited by a strong coupling and weak probe laser fields. In a weak field limit of the probe light, EIT spectrum is interpreted as functions of controllable parameters of the coupling light and temperature of the medium. The theoretical interpretation of EIT spectrum is applied to the case of Rb-85 atoms and compared with available experimental observation. Such an analytical interpretation provides quantitative parameters to control properties of the Doppler broadened EIT medium, and it is useful to find related applications.
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Ultrathin optical fibres integrated into cold atom setups are proving to be ideal building blocks for atom-photon hybrid quantum networks. Such optical nanofibres (ONF) can be used for the demonstration of nonlinear optics and quantum interference phenomena in atomic media. Here, we report on the observation of multilevel cascaded electromagnetically induced transparency (EIT) using an optical nanofibre to interface cold 87^{87}Rb atoms through the intense evanescent fields that can be achieved at ultralow probe and coupling powers. Both the probe (at 780 nm) and the coupling (at 776 nm) beams propagate through the nanofibre. The observed multipeak transparency spectra of the probe beam could offer a method for simultaneously slowing down multiple wavelengths in an optical nanofibre or for generating ONF-guided entangled beams, showing the potential of such an atom-nanofibre system for quantum information. We also demonstrate all-optical-switching in the all fibred system using the obtained EIT effect.
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We have developed an all-optical method for measuring the lifetimes of nSn S and nDn D Rydberg states and demonstrate its capabilities with measurements on a dilute cloud of ultracold 87{}^{87}Rb atoms in a cryogenic environment. The method is based on the time-resolved observation of resonant light absorption by ground state atoms and selective transfer of Rydberg atoms into the ground state at varying delay times in order to reconstruct Rydberg decay curves. Our measurements of the 87{}^{87}Rb 30S1/230S_{1/2} state indicate an increase of the lifetime at lowered environment temperatures, as expected due to decreased black body radiation. For the 38D5/238D_{5/2} state with an attractive dipole-dipole interaction, ionization and lifetime reduction due to collisional effects are observed.
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We investigate the electromagnetically induced transparency phenomenon in a four-level inverted-Y system, i.e. composite of three-level Λ and Ξ systems, using the density matrix formalism theoretically in stationary as well as moving atoms. Most of the EIT studies have taken the probe and coupling fields with wavelength matched condition to minimize the effect of Doppler-broadening. In this work we show a variation of absorption and dispersion profiles with wavelength mismatching factor because in real atoms for multilevel system the matched wavelength condition is difficult to obtain. This mismatching results in Doppler-broadening. We show that for moving atoms at the room temperature, a narrow window is also present within the wide transparency window for the probe absorption and dispersion for the mismatched cases. The group index modifies the probe propagation from super-luminal to sub-luminal within a very small detuning range near resonance. The variation of the group index with the coupling fields shows optical switching behavior when one of the coupling field is slightly detuned.
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We study a pair of probe and coupling laser pulses propagating in a Doppler broadened three-level cascade atomic medium. Influence of the coupling pulse area on the probe pulse shape is studied in a wide region of pulse duration, from micro to pico second. It is found that the needed value of coupling pulse area to establish an electromagnetically induced transparency (EIT) for the probe light, namely the probe pulse shape is unchanged, is smaller at shorter pulse duration. On the other hand, influence of Doppler broadening is more efficient in the long duration side. These results can find interesting applications in all optical switching, quantum information processing, and transmission.