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Formation and ultraslow propagation of infrared solitons in graphene under
an external magnetic field
Chunling Ding,
1
Rong Yu,
2
Jiahua Li,
3,4,a)
Xiangying Hao,
2
and Ying Wu
3,a)
1
School of Physics and Electronics, Henan University, Kaifeng 475004, People’s Republic of China
2
School of Science, Hubei Province Key Laboratory of Intelligent Robot, Wuhan Institute of Technology,
Wuhan 430073, People’s Republic of China
3
Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong University of Science and
Technology, Wuhan 430074, People’s Republic of China
4
MOE Key Laboratory of Fundamental Quantities Measurement, Wuhan 430074, People’s Republic of China
(Received 22 April 2014; accepted 4 June 2014; published online 17 June 2014)
Unusual dispersion relation of graphene nanoribbons for electrons can lead to an exceptionally
strong optical response in the infrared regime and exhibits a very good tunable frequency.
According to quantum optics and solid-material scientific principles, here we show the possibility
to generate ultraslow infrared bright and dark solitons in graphene under the action of strong
magnetic and infrared laser fields. By means of quantum-mechanical density-matrix formalism, we
derive the equations of motion that govern the nonlinear evolution of the probe-pulse envelope in
this scheme. It is found that, by properly choosing the parameters of the system, the formation and
ultraslow propagation of infrared spatial solitons originate from the balance between nonlinear
effects and the dispersion properties of the graphene under infrared excitation. Moreover, the
unique electronic properties and selection rules near the Dirac point provide more freedom for us
to study the linear and nonlinear dynamical responses of the photonics and graphene system. These
results may have potential applications in telecommunication and optical information processing.
V
C2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4883765]
I. INTRODUCTION
Graphene has been in the forefront of nanoelectronics
and condensed matter physics during the past few years. First,
graphene has intriguing electronic and optical properties due
to the fact that the linear energy-momentum dispersion of the
electrons near the Dirac point and the chiral feature of elec-
tronic states.
1–3
It has been shown that the multiple absorption
peaks and the unique selection rules of the graphene originate
from its magneto-optical properties and the peculiar thin
graphite layers.
4–7
Second, recent progress in growing high-
quality epitaxial graphene has drawn intense interest because
of the combination of unusual electronic properties and excel-
lent optical properties, these investigations provide a way to
new applications in infrared optics and photonics.
8–10
Mikhailov and Ziegler
11
have theoretically studied the nonlin-
ear electromagnetic response of classical charges in the mag-
netized graphene with linear energy dispersion. The linear and
nonlinear optical response of graphene, nonlinear frequency
conversion of terahertz (THz) surface plasmons as well as the
generation of polarization-entangled photons based on the
nonlinear optical interaction in strong magnetic field have
been studied in Refs. 12–15. In addition, it has been demon-
strated that the bilayer graphene can exhibit a giant and tuna-
ble second-order optical nonlinearity and may have potential
applications in new compact photonic and optoelectronic
devices.
16
This has made graphene important and interesting
from the practical viewpoint of quantum physics.
The interaction of optical fields with nonlinear media
gives rise to many interesting physical phenomena, including
electromagnetically induced transparency (EIT),
17–19
four-
wave mixing (FWM),
20,21
bistability (OB),
22,23
and
solitons.
24–34
In particular, optical soliton as a fundamental
nonlinear optical phenomenon has attracted a lot of interest,
which describes a special kind of electromagnetic waves that
can propagate undistorted over a long distance in nonlinear
medium. The formation of optical soliton stems from the bal-
ance between the nonlinear effects and the dispersion proper-
ties of the medium under optical excitation. Solitons have
been observed in many physical systems,
24–34
such as optical
fibers,
24,25
cold-atom media,
26–30
single molecular mag-
nets,
31,32
semiconductor quantum well,
33,34
etc.
In recent years, the optical solitons have also been investi-
gatedingraphenebothexperimentally and theoretically.
35–42
For instance, tunable dissipative soliton generation, as well as
vector multi-soliton operation and interaction in graphene ox-
ide mode-locked fiber laser have been investigated
experimentally.
35–38
McEuen et al.
39
have studied the strain
solitons in bilayer graphene by using electron microscopy to
measure with nanoscale and atomic resolution the widths,
motion, and topological structure of soliton boundaries. On the
other hand, Garcia-Vidal and collaborators
40
have theoretically
studied the formation and nonlinear propagation of subwave-
length spatial solitons in a graphene monolayer based on the
large intrinsic nonlinearity of graphene at optical frequencies.
Also, in an electrically tunable graphene metamaterial, it has
been found that the stable spatial solitary waves can propagate
along the longitudinal direction for realistic parameters.
41
Cheh and Zhao
42
have demonstrated the direct evidence of
a)
Authors to whom correspondence should be addressed. Electronic addresses:
huajia_li@163.com and yingwu2@126.com
0021-8979/2014/115(23)/234301/7/$30.00 V
C2014 AIP Publishing LLC115, 234301-1
JOURNAL OF APPLIED PHYSICS 115, 234301 (2014)
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solitons in graphene by means of molecular dynamics simula-
tions and mathematical analysis; they discovered that various
solitons emerged in the graphene flakes with two different
chiralities by cooling procedures. Especially, it has been
reported that graphene exhibits an extremely strong nonlinear
optical response in the THz or infrared regime.
44–48
As can be
seen, the graphene structures have huge potential applications
in the realm of nonlinear optics, nevertheless, largely have not
been explored yet.
Motivated by the properties aforementioned, according to
quantum optics and solid-material scientific principles, we put
forward a scheme in this paper to investigate the formation of
bright and dark spatial solitons in the infrared region and their
ultraslow propagation in graphene under an external magnetic
field. Different from these schemes discussed above,
35–42
they
either took the graphene as a saturable absorber,
35–38
or used
electron microscopy to measure the widths and motion of soli-
ton boundaries.
39
Besides, the subwavelength optical solitons
by employing a quasi-analytical model
40
and THz relativistic
spatial solitons based on the intraband optical current
41
were
discussed, and the solitons in graphene was studied by cooling
procedures.
42
While, the pulsed probe field in our scheme is
governed by nonlinear envelope equation with highly signifi-
cant nonlinearity, where the carrier frequency of the probe field
can be adjustable in the range of the infrared frequencies. The
detailed derivations of the linear and nonlinear dynamical
response of the magnetized graphene are presented by means
of a rigorous density-matrix formalism. By applying this
approach, we obtain the analytical solutions for the fundamen-
tal bright and dark solitons in the infrared range, and simultane-
ously the numerical simulation of the nonlinear wave equation
is carried out. It is clearly shown that the graphene metamaterial
can support the ultraslow propagation of infrared solitons with-
out distortion via the EIT. On the other hand, due to the unique
flexibility and strong nonlinearity of graphene, the infrared
bright and dark solitons with low absorption, high sensitivities,
and frequency adjustable can be obtained under proper condi-
tions. Our investigations about the formation and control of tun-
able solitions in graphene may have potential applications in
telecommunication and optical information processing and may
lead to substantial impact on technology.
43
The remaining of this paper is organized as follows. In
Sec. II, we mainly focus on describing the theoretical model
under consideration and deriving the governing master equa-
tions by employing the quantum-mechanical density-matrix
approach. In Sec. III, we analyze in detail the linear and nonlin-
ear dynamical properties of the photonics and graphene system.
Then, via exploiting giant nonlinear optical responses of gra-
phene, we demonstrate that the formation and ultraslow propa-
gation of infrared bright and dark solitons can be achieved by
properly choosing the practical system parameters including the
external field strength and frequency detuning under certain
conditions. Finally, our conclusions are presented in Sec. IV.
II. PHYSICAL MODEL AND GOVERNING MASTER
EQUATIONS
We consider a doped graphene system in the presence of
a strong magnetic field with four energy levels that forms a
Kconfiguration, as shown schematically in Fig. 1.
According to the peculiar selection rules of graphene, i.e.,
Djnj¼61(nis the energy quantum number) as opposed to
Dn¼61 for electrons, the chosen transitions between
Landau levels are dipole allowed. Such a system has been al-
ready used for studying the giant optical nonlinearity, nonlin-
ear frequency conversion, and generation of entangled
photons.
12–15
Optical transitions between the adjacent
Landau levels (LLs) in graphene fall into the infrared to THz
region for a magnetic field in the range of 0.01–10 T:
hxc’36 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
BðTeslaÞ
pmeV. The bichromatic electric fields
consisted of the pulsed probe field and the continuous-wave
(CW) control field can be written as ~
E1¼~
e1E1expðix1t
þi~
k1~
rÞþc:c:and ~
E23 ¼~
e2E2expðix2tþi~
k2~
rÞþ~
e3E3
expðix2tþi~
k2~
rÞþc:c:, here ~
ejand ~
kjare, respectively,
the related unit vector of the polarization field and the wave
vector with the slowly varying envelope E
j
(j¼1, 2, 3). We
assume that the probe field with right-hand circular polariza-
tion interacts with the inter-Landau-level transition j1i
$j4i, the corresponding amplitude and carrier frequency
are E
1
and x
1
, respectively. The linearly polarized CW con-
trol field ~
E23 with carrier frequency x
2
ðx3¼x2Þis applied
to drive the intra-Landau-level transition j2i$j3ivia the
r
þ
component and j3i$j4ivia the r
component (see Fig.
1), since linearly polarized light can be regarded as a linear
combination of left- and right-circularly polarized light. The
unit polarization vectors ~
e2and ~
e3denoting, respectively,
right-circular r
þ
and left-circular r
polarizations can be
expressed as ~
e2¼ð
^
xþi^
yÞ=ffiffiffi
2
pand ~
e3¼ð
^
xi^
yÞ=ffiffiffi
2
pin the
x-yplane of the graphene.
In the absence of an external optical field, the effective-
mass Hamiltonian
49–51
for a single-layer graphene (in the
x-yplane) under the magnetic field B^z(perpendicular to the
plane of graphene) can be given by
FIG. 1. (a) LLs near the Kpoint superimposed on the electronic energy
dispersion without a magnetic field E¼6vFjpj. The magnetic field
“condenses” the original states in the Dirac cone into discrete energies.
The LLs in graphene are unequally spaced: /ffiffiffi
B
p. (b) Energy level
diagram and optical transitions involved in formation and ultraslow
propagation of infrared solitons in graphene interacting with two
continuous-wave control fields 2 and 3 (with carrier frequencies x
2
and
x
3
, respectively) and a weak pulsed probe field 1 (with carrier frequency
x
1
). The states j1i;j2i;j3i,andj4icorrespond to the LLs with energy
quantum numbers n¼2, 1, 0, 1, respectively. Graphene monolayer is
a one-atom-thick monolayer of carbon atoms arranged in a hexagonal lat-
tice, which we will treat as a perfect two-dimensional (2D) crystal struc-
ture in the x-y plane.
234301-2 Ding et al. J. Appl. Phys. 115, 234301 (2014)
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218.199.86.138 On: Tue, 17 Jun 2014 22:10:37
^
H0¼vF
0^
pxi^
py00
^
pxþi^
py000
000
^
pxþi^
py
00
^
pxi^
py0
0
B
B
@1
C
C
A
;(1)
where vF¼3c0=ð2haÞ106m=s(c02:8 eV and
a¼1.42 A
˚are the nearest-neighbor hopping energy and C-C
spacing) is a band parameter (Fermi velocity), ^
~
p¼
^
~
pþe~
A=cdenotes the generalized momentum operator, ^
~
pis
the electron momentum operator, eis the electron charge,
and ~
Ais the vector potential, which is equal to (0, Bx) for a
static magnetic field. In order to consider the interaction with
the incident optical field, we need add the vector potential of
the optical field (~
Aopt ¼ic~
E=xand ~
E¼~
E1þ~
E23) to the
vector potential of the magnetic field in the generalized mo-
mentum operator ^
~
pin the Hamiltonian. The resulting inter-
action Hamiltonian can be expressed in the following form:
^
Hint ¼vF~
re
c~
Aopt:(2)
One can easily see from the above expression that, unlike the
case of an electron with a parabolic dispersion relation, there
are no higher-order terms in the vicinity of the Dirac point
for the interaction Hamiltonian (H
int
) in graphene, conse-
quently even for a relatively strong optical field, H
int
is still
linear with respect to ~
Aopt. Furthermore, the interaction
Hamiltonian does not include the momentum operator, it is
only determined by the Pauli matrix vector ~
r¼ðrx;ryÞ.
Now we can give a standard time-evolution equation for
the density matrix of Dirac electrons in graphene coupled to
the infrared laser fields by utilizing the Liouville’s equation
@^
q
@t¼i
h
^
Hint;^
q
þ^
Rð^
qÞ, here ^
Rð^
qÞindicates incoherent
relaxation which may originate from disorder, interaction
with phonons, and carrier-carrier interactions. And thus, the
density-matrix equations of motion for the coupled system
can be written as follows:
_
q11 ¼iX
1q41 iX1q14;(3)
_
q22 ¼c2q22 þiX
2q32 iX2q23;(4)
_
q33 ¼c3q33 þiX2q23 þiX
3q43 iX
2q32
iX3q34;(5)
_
q44 ¼c4q44 þiX1q14 þiX3q34 iX
1q41
iX
3q43;(6)
_
q21 ¼c2
2q21 þiX
2q31 iX1q24;(7)
_
q31 ¼ iDx2þc3
2
q31 þiX2q21 þiX
3q41
iX1q34;(8)
_
q41 ¼ iDx1þc4
2
q41 þiX1ðq11 q44Þ
þiX3q31;(9)
_
q32 ¼ iDx2þc3þc2
2
q32 þiX2ðq22 q33Þ
þiX
3q42;(10)
_
q42 ¼ iDx1þc4þc2
2
q42 þiX1q12 þiX3q32
iX2q43;(11)
_
q43 ¼ iDx1iDx2þc4þc3
2
q43 þiX1q13
þiX3ðq33 q44ÞiX
2q42;(12)
where Dx1¼ðen¼1en¼2Þ=hx1¼ðen¼1en¼0Þ=h
x3and Dx2¼ðen¼0en¼1Þ=hx2represent the corre-
sponding frequency detunings, and en¼sgnðnÞhxcffiffiffiffiffiffi
jnj
p
is the energy of the Landau level for electrons near the
Dirac point, with n¼0;61;62;…;xc¼ffiffiffi
2
pvF=lc, and lc
¼ffiffiffiffiffiffiffiffiffiffiffiffiffi
hc=eB
pimplies the magnetic length. X1¼ð
~
l41 ~
e1Þ
E1=ð2hÞ;X2¼ð
~
l32 ~
e2ÞE2=ð2hÞand X3¼ð
~
l43 ~
e3Þ
E3=ð2hÞare the corresponding one-half Rabi frequencies
with ~
lmn ¼hmj~
ljni¼ehmj~
rjni¼ ihe
enemhmjvF~
rjnibeing
the dipole matrix element for the relevant optical transition.
c
j
(j¼2, 3, 4) means the decay rate of the state jji. Moreover,
these density-matrix elements satisfy the population conser-
vation condition P4
j¼1qjj ¼1.
The evolution of the electric field is governed by the
Maxwell equation
r2~
E1
v2
F
@2~
E
@t2¼1
rv2
F
@2~
P
@t2;(13)
with the optical polarization
~
P¼Nf~
l14q41 exp½ið~
k1~
rx1tÞ
þ~
l23q32 exp½ið~
k2~
rx2tÞ
þ~
l34q43 exp½ið~
k3~
rx3tÞ þ c:c:g;
here Nand
r
being the sheet electron density of graphene
and the substrate dielectric constant, respectively. Since gra-
phene is essentially a 2D system, it makes sense to introduce
a surface 2D polarization ~
Pdetermined as an average dipole
moment per unit area rather than unit volume.
14
Under the slowly varying envelope approximation, the
Maxwell equation can be explicitly reduced into the first-
order equation. Hence we can arrive at the slowly varying
envelope equation for describing the probe field evolution,
that is,
@X1ðz;tÞ
@zþ1
vF
@X1ðz;tÞ
@t¼ij14q41 ;(14)
where j
14
is the propagation constant, given by
j14 ¼Nx1j~
l14 ~
e1j2=ð2hrvFÞ. This equation, along with
Eqs. (3)–(12), characterizes the propagation of the pulsed
probe field in our proposed system with the Rabi frequency
X
1
(z,t).
III. LINEAR AND NONLINEAR DYNAMICS AND
FORMATION OF ULTRASLOW INFRARED SOLITONS
In this section, we will analyze the linear and nonlinear
dynamical response of the system, as well as the formation
234301-3 Ding et al. J. Appl. Phys. 115, 234301 (2014)
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and propagation of bright and dark spatial solitions in the
infrared region. Under the assumption that the Rabi fre-
quency X
1
of the pulsed probe field is much smaller than that
of the control fields X
2
and X
3
, so that the Dirac electrons
are initially populated in their ground state j1iwithout deple-
tion, as a result, q11 ’1 whereas q22 ’q33 ’q44 ’0. That
is to say, we assume that state j1iis fully occupied while
states j2i;j3i, and j4iare empty; i.e., the Fermi level is
between LLs n¼2 and n¼1. Then under the condition
of perturbation expansion qmn ¼PkqðkÞ
mn, here qðkÞ
mn is the
kth-order part of qmn in terms of X
1
, it can be shown that
qð0Þ
mn ¼0ðm6¼ nÞand qðkÞ
22 ¼qðkÞ
33 ¼qðkÞ
44 ¼0. Keeping up to
the first order of the pulsed probe field and performing the
time Fourier transform of Eqs. (7)–(12) and Eq. (13),we
have the results
xþic2
2
bð1Þ
21 þX
2bð1Þ
31 K1bð0Þ
24 ¼0;(15)
xDx2þic3
2
bð1Þ
31 þX2bð1Þ
21 þX
3bð1Þ
41
K1bð0Þ
34 ¼0;(16)
xDx1þic4
2
bð1Þ
41 þK1þX3bð1Þ
31 ¼0;(17)
xDx2þic3þc2
2
bð1Þ
32 þX
3bð1Þ
42 ¼0;(18)
xDx1þic4þc2
2
bð1Þ
42 þX3bð1Þ
32 X2bð1Þ
43
þK1bð0Þ
12 ¼0;(19)
xDx1þDx2þic4þc3
2
bð1Þ
43 X
2bð1Þ
42
þK1bð0Þ
13 ¼0;(20)
and
@K1
@zix
vF
K1¼ij14bð1Þ
41 ;(21)
where b
ij
and K
1
are, respectively, the Fourier transforms of
qij and X
1
,xis the time Fourier transform variable.
Equation (21) can be solved analytically with the aid of
Eqs. (15)–(20), yielding
K1ðz;xÞ¼K1ð0;xÞexp½iKðxÞz;(22)
where
KðxÞ¼x
vFj14D1ðxÞ
DðxÞ¼K0þK1xþK2x2þOðx3Þ;
(23)
with D1ðxÞ¼jX2j2ðxþic2ÞðxDx2þic3Þand
DðxÞ¼jX2j2ðxDx1þic4ÞþjX3j2ðxþic2Þðxþic2Þ
ðxDx2þic3ÞðxDx1þic4Þ, respectively. K0¼K
ðx¼0Þ¼uþia=2 signifies the phase shift uper unit length
and the linear absorption coefficient aof the probe field.
K1¼dKðxÞ
dxjx¼0¼1
Vgrepresents the propagation group velocity
and K2¼d2KðxÞ
dx2jx¼0indicates the group-velocity dispersion
that gives the contribution to the shape variation and addi-
tional loss of field intensity.
Figure 2shows the absorption coefficient aand the rela-
tive group velocity V
g
/cof the pulsed probe field versus the
dimensionless Rabi frequency jX2j=c3. This figure (see the
dashed curve) shows that there exist some parameter ranges
where the absorption of the probe field can be almost com-
pletely suppressed owing to the destructive interference
induced by the control fields under appropriate conditions in
this graphene configuration system. It should be pointed out
that the infrared spatial soliton generated in this way propa-
gates with the ultraslow group velocity (e.g., Vg103c),
which is illustrated by the solid curve in Fig. 2. Notice that
we have obtained the linear response of the system under the
situations of the weak-field and adiabatic approximations by
neglecting higher-order terms of the probe field. In order to
form the bright and dark spatial solitons, we need to find the
effective nonlinear effect (i.e., the self-phase modulation
effect) to balance the pulse spreading and the detrimental
distortions of the pulsed probe field which is induced by the
group-velocity dispersion.
Now we turn our attention to study the nonlinear evolu-
tion of the pulsed probe field in our system. In order to
explore the formation of infrared solitons, we need to take a
trial function K1ðz;xÞ¼~
K1ðz;xÞexpðiK0zÞaccording to the
method proposed in Refs. 29–34 and substitute the trial func-
tion into the wave equation (21), we obtain
@K1ðz;xÞ
@zexpðiK0zÞ¼iðK1xþK2x2ÞK1ðz;xÞexpðiK0zÞ;
(24)
where we have replaced ~
K1ðz;xÞwith K1ðz;xÞfor the sake
of convenience.
FIG. 2. The absorption coefficient a(dashed curve) and the relative group
velocity V
g
/c(solid curve) versus the Rabi frequency jX2j=c3. The parame-
ters used for the simulations from Refs. 12 and 53 are c
3
¼310
13
s
1
,
v
F
¼10
6
m/s, jX3j¼10c3;Dx1¼Dx2¼9c3;c2¼0:05c3;c4¼c3, and
j14 ¼0:02c3lm
1
, respectively.
234301-4 Ding et al. J. Appl. Phys. 115, 234301 (2014)
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For the purpose of balancing the interaction of nonlinear
effect and the group-velocity dispersion, we need to take into
account the nonlinear polarization on the right hand side of
Eq. (14), i.e.,
ij14q41 ’ij14 qð1Þ
41 þi<;(25)
where <denotes the nonlinear term and which is given by
<¼j14qð1Þ
41 expðiK0zÞP4
j¼2jqð1Þ
j1j2, and qð1Þ
j1can be
achieved by solving Eqs. (15)–(20). After carrying out some
algebraic calculations, the solutions of the set of equations
can be written as
qð1Þ
21 ¼X
2X
3
Dð0ÞX1;
qð1Þ
31 ¼ic2X
3
Dð0ÞX1;
qð1Þ
41 ¼jX2j2þic2ðDx2ic3Þ
Dð0ÞX1;
with Dð0Þ¼jX2j2ðDx1ic4Þþic2jX3j2ic2ðDx2ic3Þ
ðDx1ic4Þ.
By means of the nonlinear polarization term, we begin
to analyze the nonlinear dynamical properties of the gra-
phene system. Carrying out the inverse Fourier transforma-
tion of Eq. (24), we can get the nonlinear wave equation for
the slow varying envelope X
1
(z,t),
i@X1ðz;tÞ
@zK2
@2X1ðz;tÞ
@t2¼WeazjX1ðz;tÞj2X1ðz;tÞ;(26)
here the absorption coefficient a¼2Im½Kð0Þ ¼ 2j14
Im½D1ð0Þ=Dð0Þ, and W¼j14D1ð0Þ½jD1ð0Þj2þjX3j2
ðjX2j2þDx2
2þc2
3Þ=½jDð0Þj2Dð0Þ is the nonlinear coeffi-
cient of the pulsed probe field.
To examine the analytical expressions of the two coeffi-
cients K
2
and W, we find that the nonlinear evolution equation
(26) has complex coefficients and generally does not allow
soliton solutions. However, in the presence of the control
fields, the absorption of the probe field can be almost com-
pletely suppressed by choosing appropriate parameters,
which results in expðaLÞ’1(Lis the length of the gra-
phene), just as exhibited in Fig. 2. Similarly, reasonable and
practical set of parameters can also be found so as to the
imaginary parts of the complex coefficients are much smaller
than their corresponding real parts for the present system, and
thus we have K2¼K2rþiK2i’K2rand W¼WrþiWi
’Wr, these have been demonstrated in Fig. 3. Therefore,
under certain conditions, Eq. (26) can be written as in the fol-
lowing form which corresponds to the standard nonlinear
Schr€
odinger equation:
i@X1ðn;gÞ
@nK2r
@2X1ðn;gÞ
@g2¼WrjX1ðn;gÞj2X1ðn;gÞ;(27)
where n¼zand g¼tz=Vg. Equation (27) admits the sol-
utions of bright and dark solitons which depends on the sign
of the product K
2r
W
r
.
25,52
That is to say, when K2rWr>0,
we can observe the bright soliton; on the contrary, we can
only observe the dark soliton in the case of K2rWr<0. The
fundamental bright and dark spatial solitons are given by
X1ðn;gÞ¼X10 sechðg=sÞexpðinjK2rj=s2Þ(28)
and
X1ðn;gÞ¼X10 tanhðg=sÞexpð2injK2rj=s2Þ;(29)
here sechðg=sÞand tanhðg=sÞrepresent the hyperbolic secant
function and hyperbolic tangent function, respectively. The
amplitude X
10
and pulse width sare arbitrary constants, we
can take s¼3:33 1014 s.
12–15
It is noted that the carrier frequency of the probe field can
be estimated approximately the same amount of the transition
frequency x41 ¼ðen¼1en¼2Þ=h¼ð ffiffiffi
2
pþ1Þxc,whichis
on the order of x41 1:86 1014 s1for graphene at the
magnetic field of the value B¼1T.
12–15
While, when the
magnetic field reaches up to 5 T, the transition frequency is
estimated to be x41 3:87 1014 s1. In the present work,
we take the magnetic field B¼3 T for example, at this
moment xc1014 s1, this is confirmed that the spatial soli-
ton is located within the infrared region. According to the nu-
merical estimate based on Refs. 12 and 53, we can take a
reasonable value for the decay rate c3¼31013 s1,and
assuming c4¼c3;c2¼0:05c3, these values depend on the
sample quality and the substrate used in the experiment.
12–15
Besides, the dipole moment between the transition j1i$j4i
in the graphene has a magnitude of the order j~
l14j
hevF=ðe4e1Þ/1=ffiffiffi
B
p. The electron concentration is
N’51012 cm2and the substrate dielectric constant is
r’4:5.
40–42,54
In Fig. 4, we show the results of numerical
simulation on the soliton waveform jX1=X10j2versus the
dimensionless time g=sand distance n=Lwith the full com-
plex coefficients by taking Eq. (28) ðK2rWr>0Þand Eq. (29)
ðK2rWr<0Þas initial conditions. It can be seen from Fig. 4
that in this case the bright and dark spatial solitons remain
fairly stable during propagation, which are mainly generated
from the balance between the group-velocity dispersion
described by the coefficient K
2
and Kerr-type self-phase mod-
ulation effect. Thus the results of numerical simulations give
FIG. 3. The ratios K
2i
/K
2r
(dashed curve) and W
i
/W
r
(solid curve) versus the
Rabi frequency jX2j=c3. The system parameters used for the simulations are
the same as those in Fig. 2.
234301-5 Ding et al. J. Appl. Phys. 115, 234301 (2014)
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extremely good agreement with the exact soliton solutions
calculated in Eqs. (28) and (29). In view of rapid advances in
graphene material, we believe that the formation of ultraslow
infrared solitons will be highly accessible in experiments in
near future.
IV. CONCLUSIONS
In conclusion, we have theoretically studied the forma-
tion and control of the spatial solitons in a magnetized gra-
phene driven by tunable infrared picosecond laser pulses by
using density-matrix method and perturbation theory. The
infrared bright and dark solitons were obtained owing to the
balance between nonlinear effects and the dispersion proper-
ties of the graphene under infrared excitation, which pre-
vented devastating collapse. Moreover, due to the remarkable
electronic properties and selection rules near the Dirac point,
the formation and ultraslow propagation of the infrared soli-
tons can be controlled by adjusting the practical system pa-
rameters including the external field strength and detuning
under appropriate conditions. Such ultraslow infrared solitons
in graphene may provide a new possibility for designing
high-fidelity optical delay lines and optical buffers, as well as
compact all-fiber switchable laser. Also, the scheme under
study may have important practical applications in telecom-
munication and optical information processing and may cause
significant effect on technological applications due to unusual
electronic and optical properties stemming from linear, mass-
less dispersion of electrons near the Dirac point and the chiral
character of electron states.
ACKNOWLEDGMENTS
The authors express their gratitude to the referee of the
paper for his/her fruitful advice and comment, which
significantly improved the paper. This research was supported
in part by the National Natural Science Foundation (NNSF) of
China (Grant Nos. 11375067, 11275074, and 11104210), by
the National Basic Research Program of China (Contract No.
2012CB922103), and by the Doctoral Foundation of the
Ministry of Education of China under Grant No.
20134103120005. Also, we would like to thank Professor
Xiaoxue Yang for her encouragement and helpful discussion.
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