We show that the simultaneous application of a copropagating saturating pump and a counterpropagating coherent beam can be used to burn a narrow spectral hole within the absorption line of the optical transition in a Doppler-broadened medium. The large index of refraction of this hole slows down a light pulse by a factor of about 104. In addition, we propose a method to create two-color slow light pulses with simultaneous gain by employing a bichromatic field to saturate the medium

Gao, Jin-Yue

He, Qiong-Yi

Kuang, Shang-Qi

Wei, Xiao-Gang

Wu, Jin-Hui

Zhang, Bing

University of Queensland eSpace

Slow light by coherent hole burnings
Qiong-Yi He,1,2 Bing Zhang,1 Xiao-Gang Wei,1 Jin-Hui Wu,1 Shang-Qi Kuang,1 and Jin-Yue Gao1,*
1Key Laboratory of Coherent Light and Atomic and Molecular Spectroscopy of Ministry of Education, College of Physics,
Jilin University, Changchun 130023, People’s Republic of China
2ARC Centre of Excellence for Atom-Optics, School of Physical Sciences, The University of Queensland, Brisbane,
Queensland 4072, Australia
Received 22 June 2007; published 18 June 2008
We show that the simultaneous application of a copropagating saturating pump and a counterpropagating
coherent beam can be used to burn a narrow spectral hole within the absorption line of the optical transition in
a Doppler-broadened medium. The large index of refraction of this hole slows down a light pulse by a factor
of about 104. In addition, we propose a method to create two-color slow light pulses with simultaneous gain by
employing a bichromatic field to saturate the medium.
DOI: 10.1103/PhysRevA.77.063827 PACS numbers: 42.50.Gy, 42.65.k, 42.30.Rx
I. INTRODUCTION
In dispersive media, the reduction of the optical group
velocity by several orders of magnitude has attracted a great
deal of interest 1–6. There are many applications that can
benefit from light with slow group velocities, for example,
controllable optical delay lines 7, low-light-level nonlinear
optics 8, optical communication, and quantum-information
storage. Several methods have been used to achieve slow
light: electromagnetically induced transparency EIT
9–12, population oscillations 13, nonlinear magneto-
optics 3, gain resonances 14,15, and photonic lattices
16. In particular, Agarwal and Dey 17 have proposed a
Doppler-broadened two-level system to produce slow light
using optical hole burning OHB. Shakhmuratov et al. 18
extended this work to a persistent spectral hole-burning
PSHB technique that can be realized in many solid materi-
als.
Considering both EIT and OHB, Dong and Gao 19 pro-
posed a coherent hole-burning CHB technique that allows
the simultaneous observation of an EIT window and four
hole burnings when two laser beams one saturating and one
coupling drive a Doppler-broadened atomic medium in the
ladder configuration. Wu et al. 20,21 extended the CHB
theory into a -type Doppler-broadened atomic system and
observed better CHBs because of matched transition wave-
lengths.
In this paper, we propose an approach to achieve slow-
down of a light pulse with CHB. We burn a narrow spectral
hole in the Doppler-broadened absorption line by the simul-
taneous application of a copropagating saturating pump and
a counterpropagating coherent beam. Using this method, we
show that the group velocity in the atomic vapor is reduced
by several orders of magnitude, a larger slowdown than that
induced by general hole burning under the same conditions.
We can also obtain a steeper slope of the refractive index
with larger transmission when the probe pulse is slightly de-
tuned from resonance. An important advantage of using CHB
is that two narrow propagating pulses accompanied by am-
plification can be retarded when a bichromatic laser field is
used as the saturating pump.
II. THE MODEL AND EQUATIONS
The -type system under consideration is shown in Fig. 1
for two different excitation schemes. In the first scheme,
shown in Fig. 1a, levels 2 and 3 are coupled by an
intense laser with frequency c and Rabi frequency c. A
weak probe field with frequency p and Rabi frequency p
detects the absorption on transition 1↔ 3. Another laser
with frequency s and Rabi frequency s also acts on tran-
sition 1↔ 3, and is used to saturate the medium. c
=32−c, s=31−s, and p=31−p are detunings of the
three fields. 31 and 32 denote the spontaneous decay rates
from level 3 to levels 1 and 2. 21 designates the popu-
lation damping rate of levels 1 and 2.
Based on the semiclassical theory, using the standard
density-matrix formalism with the dipole approximation and
the rotating wave approximation, the interaction Hamiltonian
HI for this system is
HI = s − c22 + s33 − s13 +c23 + c.c. .
1
For this scheme, the detailed calculation of the density-
matrix equations in the interaction picture and the generation
or degeneracy of CHBs under the dressed-state representa-
*Corresponding author. jygao@mail.jlu.edu.cn
(b)
∆
s
∆c
Γ
21
ω
c
ω
p
ω
s
Γ
31
∆
p
1>
3>
2>
Γ
32
ω
s2
ω
s1
∆c
Γ
21
δ
ω
cωp
Γ
31
∆
p
1>
3>
2>
Γ
32
δ
(a)
FIG. 1. Color online Energy level diagram of a Doppler-
broadened -type system driven by probe p, coupling c, and a
saturating s or b bichromatic saturating s1,s2 beams.
PHYSICAL REVIEW A 77, 063827 2008
1050-2947/2008/776/0638275 ©2008 The American Physical Society063827-1
tion has been explained in our previous work 20,21.
Here, we emphasize the method of calculation for the
second excitation scheme, shown in Fig. 1b, where we ex-
tend the CHB technique by employing a bichromatic linearly
polarized laser field to excite the atoms to saturation,
Et =
1
2
Es1eit + Es2e−ite−i31t + c.c. 2
Es1 and Es2 are the amplitudes of the two frequency compo-
nents s1 and s2 of the bichromatic field. s1s2=s1s2
−31= + − is the corresponding detuning. 2 is the fre-
quency difference between the two laser field components.
Therefore, the master equation for the density-matrix opera-
tors should be
˙11 = is1
 e−it +s2
 eit31 − is1eit +s2e−it13
+ 3133 + s22 − 11 ,
˙22 = ic
32 −c23 + 3233 + s11 − 22 ,
˙12 = is − c − 	1212 + is32 − ic
13,
˙13 = is − 	1313 + is1eit +s2e−it33 − 11 − ic12,
˙23 = ic − 	2323 + ic33 − 22 − is1eit +s2e−it21,
11 + 22 + 33 = 1,
ij =  ji

. 3
Here, s1=Es1
13 /2, s2=Es2
13 /2, and c=Ec
23 /2
are the Rabi frequencies, where 
ij are the electric dipole
moments. 	ij refers to the rate of coherence decay between
levels i and j: 	12=	21=s, 	13=	31=	23=	32= 12 31
+32+s.
In order to solve these equations, we expand the density-
matrix elements  jk in a Fourier series as
 jk = 
n=−
+
Xk
neint j,k = 1,2,3 , 4
where Xk
n represent the slowly varying amplitudes and are
needed to determine the absorption spectra, and Xkk=1–8
n
= 31,13,21,12,32,23,11,22T 22.
According to the linear response theory, the steady-state
absorption and refraction spectrum of the weak probe laser
can be written as
Ap = Re	

0

lim
t→
P−t + ,P+teip , 5
np = Im	

0

lim
t→
P−t + ,P+teip , 6
where P−=
3131 and P+=
1313 are atomic polariza-
tion operators with dipole moments 
31 and 
13 on transition
1↔ 3.
Following the method of calculations for the bichromatic
manipulation of absorption spectrum 23, we can get the
refractive index spectra of the probe laser. To consider the
Doppler frequency shift, we have to substitute p, s, and c
with p+31v /c, s+31v /c, and c−32v /c, where + −
denotes that the saturating coupling laser is copropagating
counterpropagating with the probe laser. If the number of
atoms per unit volume with velocity v is Nvdv, then the
absorption and refraction spectra of the probe laser read
Ap = 

−

Ap,vNvdv = 

−

Ap,v
N0
vp
e−v
2/vp
2
dv ,
7
np = 

−

np,vNvdv = 

−

np,v
N0
vp
e−v
2/vp
2
dv ,
8
where N0 is the total number of atoms, and vp=2kTm =2RTM
represents the most probable atomic velocity. The Doppler
width of the probe absorption spectrum is given by D
=
31
c
8kT ln 2
m
=
231
c
2RT ln 2
M =2ln 2
31vp
c
.
Of particular interest to us is the group velocity of pulse
light, which is determined by the relation vg=c /ng,where ng
is the group index: ng=np+
np
p
.
III. NUMERICAL RESULTS AND DISCUSSION
A. The first scheme [Fig. 1(a)]
The Doppler-broadened probe absorption, refractive in-
dex, and group index as functions of the frequency detuning
of probe field are given in Fig. 2. At resonance, there exists a
rather narrow and deep hole. The other two side CHBs are
asymmetric both in position and in depth. This is consistent
with our previous theoretical analysis 20,21,23, which pre-
dicts that CHBs should be located at p=s and p= c
+5s3c+s2+8c2. The steep dispersion of the cen-
tral hole creating the large group index at the same frequency
will cause slow light with large slowdown factor. We have
also shown the behavior in the region of resonance with dif-
ferent intensity of s 4, 3, 2.5, and 2 MHz in Figs.
2a–2c. There is a coherent contribution to this hole
from the response for the coupling and probe fields and an
incoherent contribution from the hole burning.
If we take out the pump s and change the propagation
direction of the coupling field c in order to eliminate the
Doppler broadening, this scheme becomes the typical EIT. It
is well known that EIT allows us to obtain a steeper slope of
the refractive index and thereby a large group index; how-
ever, it is usually necessary to cool the medium to very low
temperatures. We need a technique, such as OHB, in which it
is possible to provide a high slowdown factor of the group
velocity at room temperature. Here, we propose another op-
tion, using CHB technology, to create slow light at room
temperature. It is well known that the Doppler-broadening
effect usually degrades the slow light effect, while in our
approach, it can instead be used to spectrally tune the slow
light propagation.
For the detuning value of s=20 MHz and c
=0, 50, 100, 150 MHz Figs. 2a–2c, we can find
HE et al. PHYSICAL REVIEW A 77, 063827 2008
063827-2
that the CHB at resonance moves to p=20 MHz. At the
same time, a larger c can be chosen to obtain a steeper
slope of the refractive index and thereby a larger group index
with higher transmission. This phenomenon can be under-
stood as follows. On the one hand, the Maxwell speed dis-
tribution shows that the atoms with the most probable veloc-
ity occupy the large scale within the whole group. In our
case, we assume that the most probable atomic velocity is
vp=250 m /s. On the other hand, two groups of atoms with
velocity v1,2= c−3sc+s2+8c2 c4s at level 1
can be excited to saturation due to the presence of the cou-
pling field psc, s=−c=1 20,21. So, for the
central CHB, we can adjust the appropriate parameters to
make v1,2 both close to vp so as to excite more atoms to
saturation and thereby obtain a steeper slope of the refractive
index.
Fixing the parameters s=20 MHz, c=40 MHz, and
s=794.969 nm for the 87Rb D1 line, we can give detailed
values of v1,2=11,−35; 24,−28; 41,−25; and 58,−22 m /s
corresponding to different c=0, 50, 100, and 150 MHz. It is
clear that v1,2=58,−22 resulting from c=150 MHz are
closest of these to vp. If c is increased even more, for ex-
ample, c=200 MHz, we can get v12=76,−21, which are
closer to vp so that the group velocity continues to increase.
But we should note that it is better not to increase c too
much because the spectral detuning of the coupling field de-
creases the width of the spectrum of the transparent probe
pulse, which is shown by the blue curve in Fig. 2c.
It is clear that the peak value of the group index induced
by CHB is 25 times larger than that obtained by general
OHB under the same conditions, as shown in Fig. 3. Here,
we give the typical inhomogeneous hole burning, which be-
comes deeper and wider with increasing intensity of s.
B. The second scheme [Fig. 1(b)]
For this excitation scheme, more interesting phenomena
can be observed: By reducing , we can find a deep and
narrow hole at resonance Fig. 4a, dashed curve in the
absorption profile that is similar to the EIT window solid
curve. It is clearly shown in the blowup that the two borders
of the hole at s=s s=0, =1 MHz go deep below
zero, that is, gain appears, because of the strong saturation
excited by the two field components of the bichromatic laser.
The slopes of the refractive index at the edges of the hole are
especially steep Fig. 4b, which can produce two narrow
slow light pulses simultaneously Fig. 4c. When the detun-
ings are changed to s=20 MHz and c=150 MHz, the
FIG. 2. Color online a Absorption, b re-
fraction, and c group index variation with de-
tuning of the probe field in the presence of a co-
propagating saturating field s=4 MHz and a
counterpropagating coupling field c=40 MHz.
a− c show blowups of a–c with different
s. a− c show the same as the black curve
in a− c with different detunings.
SLOW LIGHT BY COHERENT HOLE BURNINGS PHYSICAL REVIEW A 77, 063827 2008
063827-3
hole at resonance moves to p=s=20 MHz Figs.
4a–4c. The transparency window becomes wider and
the edges of the hole at s=s s=20 MHz, 
=1 MHz become deeper; gain increases. The two slopes of
the refractive index at the edges of the hole are steeper and
thereby produce larger group index value. This method offers
the possibility of observing the reduced propagation speed of
a two-color field accompanied by amplification. Also, we can
conveniently control the frequency and slowdown factor of
the probe field by modulating the detunings of both coupling
and saturating beams.
This figure shows clearly that the slow light induced by
bichromatic coherent hole burning dashed curves has great
advantages: higher slowdown factor with gain and a two-
color slow light field. On the one hand, the two retarded
propagation pulses with gain have many advantages in high-
resolution spectroscopy and frequency stabilization. On the
other hand, several holes can be burnt in different parts of the
inhomogeneous absorption line and thereby several probe
pulses with frequencies coinciding with the frequency posi-
tions of these holes can be slowed down simultaneously. This
can be used for application in the construction of multichan-
nel optical storage devices.
IV. CONCLUSION
In summary, the coherent hole burning in a Doppler-
broadened -type system enables us to achieve slow light.
When the system is excited by a bichromatic laser field, we
can slow down two narrow pulses with large gain simulta-
neously. This can be demonstrated experimentally because
the detailed analyses in our paper are carried out for the
relevant hyperfine levels of the 87Rb D1 line, 31
=794.969 nm and 32=794.983 nm. If we can extend this
CHB technique from an atomic vapor with slow decay time
and transient spectral hole-burning properties into solid ma-
terials exhibiting fast decay time and PSHB properties, we
believe that the present approach has excellent potential for
applications in controlling the width of the spectral holes and
the optical group velocity of the propagation pulse.
ACKNOWLEDGMENTS
We would like to acknowledge support from the NSFC
Grant No. 10334010 and the National Basic Research Pro-
gram Grant No. 2006CB921103. Q.Y.H. also wishes to ac-
knowledge support from the ARC Centre of Excellence for
Quantum-Atom Optics.
FIG. 3. Color online a Absorption, b refraction, and c
group index variation with detuning of the probe field in the pres-
ence of a copropagating pump field s. The common parameters for
87Rb vapor are chosen as Doppler width parameter D=500 MHz,
density N=21012 /cm3, 31=32=6 MHz, 21=10 KHz, and
s=0.
FIG. 4. Color online a Absorption, b refractivion, and c
group index variation with detuning of the probe field. Comparison
of EIT black, solid with c=30 MHz and coherent hole burning
red, dashed induced by a copropagating bichromatic saturating
field s1=s2=4 MHz, =1 MHz, and a counterpropagating cou-
pling field c=40 MHz. The inset shows the blowup. a− c
show the same as a–c but with different detunings s
=20 MHz and c=150 MHz.
HE et al. PHYSICAL REVIEW A 77, 063827 2008
063827-4
1 L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature
London 397, 594 1999.
2 M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R.
Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O.
Scully, Phys. Rev. Lett. 82, 5229 1999.
3 D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yash-
chuk, Phys. Rev. Lett. 83, 1767 1999.
4 C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, Nature
London 409, 490 2001.
5 D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and
M. D. Lukin, Phys. Rev. Lett. 86, 783 2001.
6 A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y.
Chiao, Phys. Rev. Lett. 86, 3925 2001.
7 R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner,
Phys. Rev. A 71, 023801 2005.
8 D. A. Braje, V. Balic, G. Y. Yin, and S. E. Harris, Phys. Rev. A
68, 041801R 2003.
9 O. Kocharovskaya and Ya. I. Khanin, Zh. Eksp. Teor. Fiz. 90,
1610 1986 Sov. Phys. JETP 63, 945 1986.
10 K. J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev. Lett.
66, 2593 1991.
11 J. E. Field, K. H. Hahn, and S. E. Harris, Phys. Rev. Lett. 67,
3062 1991.
12 S. E. Harris, Phys. Today 50 7, 36 1997.
13 M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Phys. Rev.
Lett. 90, 113903 2003; Science 301, 200 2003.
14 J. E. Sharping, Y. Okawachi, and A. L. Gaeta, Opt. Express
13, 6092 2005.
15 Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Sch-
weinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, Phys.
Rev. Lett. 94, 153902 2005.
16 X. Zhao, P. Palinginis, B. Pesala, C. Chang-Hasnain, and P.
Hemmer, Opt. Express 13, 7899 2005.
17 G. S. Agarwal and Tarak Nath Dey, Phys. Rev. A 68, 063816
2003; 73, 043809 2006.
18 R. N. Shakhmuratov, A. Rebane, P. Megret, and J. Odeurs,
Phys. Rev. A 71, 053811 2005.
19 P. Dong and J. Y. Gao, Phys. Lett. A 265, 52 2000.
20 J. H. Wu, X. G. Wei, D. F. Wang, Y. Chen, and J. Y. Gao, J.
Opt. B: Quantum Semiclassical Opt. 6, 54 2004.
21 X. G. Wei, J. H. Wu, H. H. Wang, A. Li, Z. H. Kang, Y. Jiang,
and J. Y. Gao, Phys. Rev. A 74, 063820 2006.
22 This is the typical method that is often used to deal with a
system driven by a bichromatic field, for example, the work
done by X. M. Hu, Opt. Commun. 249, 543 2005; 260, 196
2006.
23 Q. Y. He, X. G. Wei, J. H. Wu, B. Zhang, and J. Y. Gao, Opt.
Commun. 28111, 3137 2008. In this paper we introduced
the detailed calculations of the absorption spectrum of the me-
dium excited by a bichromatic laser field in a -type Doppler-
broadened system.
SLOW LIGHT BY COHERENT HOLE BURNINGS PHYSICAL REVIEW A 77, 063827 2008
063827-5


Slow light by coherent hole burnings

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Slow light by coherent hole burnings

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