J. Mod. Phys., 2010, 1, 276-280
doi:10.4236/jmp.2010.14038 Published Online October 2010 (http://www.SciRP.org/journal/jmp)
Observation of Superluminal in Doppler Broadened
Two-Level Atomic Systems in Magnetic Field
Shuangqiang Liu, Yundong Zhang*, Hao Wu, Ping Yuan
Institute of Opto-Electronics an d National Key Laboratory of Tunable Laser Tech nology,
Harbin Institute of Technology, Harbin, China
Received June 23, 2010; revised July 23, 2010; acc epted July 28, 2010
Abstract
A novel method to control the group velocity of light propagation in a two-level atomic system without addi-
tional optical field is proposed. Numerical result and experimental data shows that by changing the magnetic
field intensity and vapor temperature, the group velocity of probe light can be controlled in an appropriate
region.
Keywords: Multiphoton Processes, Atom Optics, Hyperfine Structure, Zeeman Effect
1. Introduction
Recently, the precise control of group velocity of light
propagation has made many progresses in both sublu-
minal and superluminal. Slow light propagation can be
achieved by using the techniques of electromagnetically
induced transparency (EIT) [1-4], coherent population
oscillations (CPO) [5-6], coherent hole burning (CHB)
[7-8], two-wave mixing [9], stimulated Brillouin scatter-
ing (SBS) [10], stimulated Raman scattering (SRS) [11],
and so on. In superluminal area Brillouin showed theo-
retically that inside an absorption line the dispersion is
anomalous, which resulted in a group velocity faster than
c in 1960 [12], and negative group velocity of –c/(300 +
30) in Cs atom vapor was found in 2000 [13]. After their
works Bigelow et al. realized the transition from super-
luminal to subluminal propagation in alexandrite crystal
at room temperature [6]. Agarwal and Dey put forward
their theoretical discussion about sub- and superluminal
propagations in saturated and reverse saturated absorp-
tion (RSA) materials [14]. By using this method, our lab
had observed superluminal in C60 solution [15] and er-
bium-doped fiber [16]. These methods show the possibil-
ity to control the group velocity of light, however, there
are some drawbacks in these schemes, in which strong
couple field is needed (EIT, CHB) or some very special
levels structure is required (CPO, RSA), or which only
works in fiber (SBS, SRS), what make these schemes
difficult to be realized in different situations.
In this paper we give a proposal which can be used to
change the group velocity of probe field by controlling
the intensity of magnetic field and atomic vapor tem-
perature. It is well known that energy levels will split in
magnetic field because of Zeeman Effect, so the absorp-
tion characteristic of the medium will change, what leads
to the alteration of the group velocity of the probe light.
We theoretically put forward the variation of the group
velocity of the probe light when it propagates though an
atomic vapor under magnetic field, and observed the
time advance experimentally. Our results should be
helpful to realize light propagation in Doppler-broadened
atomic systems.
2. Theory and Equations
It is assumed that the magnetic field is homogeneous
within the atomic vapor and that the incident linearly
polarized radiation is transverse to the magnetic field,
therefore the polarization can be resolved into equal am-
plitude left-hand and right-hand circularly polarized com-
ponents. For an atom with a total electron angular mo-
mentum J, nuclear spin I, total angular momentum F = I
+ J, and projection of the total angular momentum along
the direction of the external magnetic field M, the Ham-
iltonian matrix elements (F,F’) for each value of M are
[17]

S. Q. LIU ET AL.
277


  
1/2
1
', '1121212'1
11'
'0
MJ I
FjI
IJFMHIJF MEF FBggJJJFF
JJFF
FIFMM


 




 



 
(1)
here the first term represents the hyperfine interaction
energy, the second represents the external magnetic
energy, μ is the Bohr magneton, B is the external mag-
netic field,
j
g
and
I
g
are the gyromagnetic factors
for the total orbital angular momentum and nuclear
spin momentum, respectively. The matrix element de-
pendence on M is contained within a 3-j symbol (en-
closed by the brackets), and the matrix element depend-
ence on F is related to a 6-j symbol (enclosed by the
braces).






 
1.51 211
1
2221221
F
KKII JJ
EhAKhB II JJ

 
 
(2)
with

111KFF JJ IIand A and B are
the magnetic dipole and electric quadruple constants for
the energy level of interest, respectively.
In the following, the response of the atomic system to
the applied field is to be discussed by defining the sus-
ceptibility

. Taken into account the contribution from
each hyperfine component, the susceptibility

can be
expressed as:
 


 
2
'' 00
22
,' '
''
32 '
,,'' 21
4
1'
212'1 1' '1'
FF MMeD
nFM F M
FF MM
In efJJ
FMF MiJ
mv
FFJIF
NF FW
MMFJ
 






 





(3)
Here ,' '
()
FM F M
W

is the plasma dispersion function,




2
,' '
,''
2/
x
FM F M
DFMFM
ie
Wdx
xIn i
 




 (4)
with

is the wavelength and

is the lifetime of the
excited level. (' )
f
JJ is absorption oscillator
strength from the ground state to excited state.
D


is
the Doppler width, which can be calculated by
2
22
DkT
In cM

  (5)
n
N is the population density of the ground-state Zee-
man level M (assuming that the optical pumping is neg-
ligible), there is
0
exp
exp
FM
nFM
h
kT
NN
h
kT











(6)
0
N is the atomic density of 1/2 6S, given by [18].
The group velocity and the group refractive index can
be given by:
 
Re
12Re 2
gg
cc
n



 


(7)
3. Numerical Results and Discussion
Basing on the above formulas in the previous section, we
take Cs 2
D transition (1/23/2 6( 4)6SF P ) for exam-
ple, and summarize our results for the steady state be-
havior of the system. Because the Doppler width (about
379 MHz at 303 K) is far larger than the splitting be-
tween the sub-levels of 3/2
6P at room temperature, the
sub-levels of 3/2
6P (F = 3, 4, 5) can’t be differentiated
in this situation.
For the transition 1/23/2 6( 4)6SF P , based on the
selection rule for electric dipole transition (0,F 1,

1m

 ), there are twenty-four left-hand and right-hand
transition, respectively. By using Equation 3 and taking
into account the contribution from each hyperfine com-
ponent, we show the behaviors of real and imaginary
parts of the susceptibility in Figure 1(a) and Figure 1(b),
respectively.
It’s well known that the imaginary and real part of the
susceptibility reflect the absorption and dispersive prop-
erty of the medium, respectively. As is shown in Figure

S. Q. LIU ET AL.
278
(a)
(b)
Figure 1. The real (a) and imaginary (b) parts of suscepti-
bility versus probe field detuning. Other parameters of
above curves are T = 303 K, γ/2π = 5.22 MHz.
1(a) that when the intensity of magnetic field B increases,
the absorption on resonance reduces and the peak of the
curve has an obvious excursion. When the intensity of
magnetic field gets strong enough, energy level degen-
eration takes place at 3/2
6P because the Zeeman split-
ting is far larger than the Doppler width, so additional
absorption peaks will appear. That will lead to an
amusive result, for the probe light of certain frequency
(see green pane in Figure 1(a)), light propagation can
change from slow light to superluminal when the mag-
netic field changes, and when the magnetic field gets
larger, the propagation changes back to slow light. The
dispersive properties of the medium are shown in Figure
1(a), from which we can see that the slope of the disper-
sion changes at different magnetic field, and the group
refractive index on resonance is planned in Figure 2 by
using Equation (7).
Figure 2. The group refractive index on resonance versus
intensity of magnetic field. The red line is the calculated
result at cell temperature T = 302 K while the black line is
the result at T = 297 K.
The curves in Figure 2 show that when magnetic field
increases the group refractive index on resonance stays
negative but its absolute value reduces and tends to be a
constant when magnetic field gets large enough (about
250 G), while the cell absorption reduces. Figure 2 also
shows the vapor temperature is an important parameter
to this system, the higher the temperature, the faster the
light. Consequently, the group velocity of the probe light
can be controlled by changing the intensity of magnetic
field B and vapor temperature T.
4. Experimental Setup
In the experiment, the correlative energy levels were Cs
1/23/2
6( 4)6SF P

, a single laser (locked to the ab-
sorption peak of Cs 1/2 3/2
6( 4)6SF P and its band-
width is 2MHz) was employed as a light source, and the
electro-optic modulator was used to modulate the laser
beam into amplitude modulated signal 0(1cos( ))
I
Im t .
A beam splitter divided the modulated optical field into
two parts, one part (5%) used as a reference and the other
as a signal. The signal light went through a Cs atomic
vapor of 3cm before it was detected. The reference ran
through the same cell without Cs in it in order to com-
pensate the dispersion from the surface of cell. The
transmitted signal and reference were received by the
detectors, and the fed into a two-channel digital oscillator
for comparison. Finally, we got the time advancement or
delay of the signal light relative to the reference through
data processing.
When the input power of the signal was 15 mW,
modulation frequency was 1.5 MHz, we observed dis-
tinct time advancement of the signal relative to the ref-
erence light through Cs cell and show it in Figure 4. By

S. Q. LIU ET AL.
279
Figure 3. Experiment setup for Cs superluminal observation.
Figure 4. The waveform of a 1.5 MHz amplitude modulated
signal propagating through 3 cm empty cell (black line), 3
cm Cs cell without magnetic field on (blue line), and 3 cm
cell in 300 G magnetic field (red line) at room temperature
T = 293 K.
Figure 5. Observed time advancement (solid dot) and theo-
retically calculated result (red line) as a function of the in-
tensity of magnetic field at room temperature T = 293 K.
using (1) /
adv g
tLnc , we got 299
g
n (B = 0) and
179
g
n (B = 300 G). Figure 5 displays the time ad-
vancement as a function of the magnetic field, form which
Figure 6. Observed time advancement (solid dot) and theo-
retically calculated result (black line) as a function of the cell
temperature at the intensity of magnetic field B = 200 G.
we can see clearly that by changing the magnetic field,
the time advancement can be controlled in an appropriate
region. Figure 6 shows the pulse passes faster because
more atoms are provided to the system when the cell
temperature is higher.
5. Conclusion
In conclusion, superluminal in Doppler broadened two
level atomic systems under magnetic field was theoreti-
cally brought forward, and observed in experiment. It is
shown that both the group refractive index and vapor
absorption depend on the magnetic field intensity and
vapor temperature. By only changing those parameters,
the group velocity of light can be controlled in an appro-
priate region, and our work should be helpful to realize
superluminal propagation in various Doppler-broadened
atomic systems.
6. Acknowledgements
The research is supported by the National Natural Sci-
ence Foundation of China under Grant Nos. 60272075,
60478014 and 60878006, the National High Technology
Research and Development Program (“863” Program) of
China under Grant No. 2007AA12Z112.
7. References
[1] S. E. Harris, “Electromagnetically Induced Transpar-
ency,” Physics Today, Vol. 50, No. 7, 1997 pp. 36-42.
[2] J. P. Marangos, “Topical Review Electromagnetically
Induced Transparency,” Journal of Modern Optics, Vol.
45, No. 3, 1998, pp. 471-503.
[3] J. G. Banacloche, Y. Li, S. Jin and M. Xiao, “Observation

S. Q. LIU ET AL.
280
of an Electromagnetically Induced Change of Absorption
in Multilevel Rubidium Atoms,” Physical Review A, Vol.
51, No. 3, 1997, pp. R1754-R1757.
[4] M. Fleischhauer, A. Imamoglu and J. P. Marangos,
“Electromagnetically Induced Transparency: Optics in
Coherent Media,” Reviews of Modern Physics, Vol. 77,
No. 2, 2005, pp. 633-678.
[5] M. S. Bigelow, N. N. Lepeshkin and R. W. Boyd, “Ob-
servation of Ultraslow Light in a Ruby Crystal at Room
Temperature,” Physical Review Letters, Vol. 90, No. 11,
2003, p. 113903.
[6] M. S. Bigelow, N. N. Lepeshkin and R. W. Boyd, “Su-
perluminal and Slow Light Propagation in a Room-Tem-
perature Solid,” Science, Vol. 301, No. 5630, 2003, pp.
200-202.
[7] P. Dong and J. Y. Gao, “Appearance and Disappearance
of Hole-Burrning behind an Electromanetically Induced
Transparency Window,” Physical Letter A, Vol. 265, No.
1-2, 2000, pp. 52-57.
[8] J. H. Wu, X. G. Wei, D. F. Wang, Y. Chen and J. Y. Gao,
“Coherent Hole-Burning Phenomenon in a Doppler Braod-
ened Three-Level-Type Atomic System,” Journal of Op-
tics B, Vol. 6, No. 1, 2004, pp. 54-58.
[9] E. Podivilov, B. Sturman, A. Shumelyuk and S. Odoulov,
“Light Pulse Slowing down up to 0.025 Cm/S by Pho-
torefractive Two-Wave Coupling,” Physical Review Let-
ters, Vol. 91, No. 8, 2003, p. 83902.
[10] K. Y. Song, M. G. Herraez and L. Thevenaz, “Gain-As-
sisted Pulse Advancement Using Single and Double Bril-
louin Gain Peaks in Optical Fibers,” Optic Express, Vol.
13, No. 24, 2005, pp. 9758-9765.
[11] J. Q. Liang, M. Katsuragawa, F. L. Kien and K. Hakuta,
“Slow Light Produced by Stimulated Raman Scattering in
Solid Hydrogen,” Physical Review A, Vol. 65, No. 3,
2002, p. 31801.
[12] L. Brillouin, “Wave Propagation and Group Velocity,”
Academic Press, New York, 1960.
[13] L. J. Wang, A. Kuzmich and A. Dogariu, “Gain-Assisted
Superluminal Light Propagation,” Nature, Vol. 406, No.
6793, 2000, pp. 277-283.
[14] G. S. Agarwal and T. N. Dey, “Sub- and Superluminal
Propagation of Intense Pulses in Media with Saturated
and Reverse Absorption,” Physical Review Letters, Vol.
92, No. 20, 2004, p. 203901.
[15] H. Wang, Y. D. Zhang, N. Wang, W. L. Yan, H. Tian, W.
Qiu and P. Yuan, “Observation of Superluminal Propaga-
tion at Negative Group Velocity in C60 Solution,” Ap-
plied Physics Letters, Vol. 90, No. 12, 2007, p. 121107.
[16] H. Wang, Y. D. Zhang, L. Ma, Z. S. He, H. Tian and P.
Yuan, “Experimental Observation of Negative Group Veloc-
ity in C-60 PMMA,” OPTIK, Vol. 120, No. 13, 2009, pp.
673-675.
[17] B. Yin and T. M. Shay, “Theoretical Model for a Faraday
Anomalous Dispersion Optical Filter,” Optical Letters,
Vol. 16, No. 20, 1991, pp. 1617-1619.
[18] Daniel A. Steck, Cesium D Line Data, 1998.
http://stech.us/alkalidata