Slow light propagation and bistable switching in a graphene under an external magnetic field

Abstract

In this letter, we show the possibility of controlling the optical bistability and group index switching in graphene under the action of strong magnetic and infrared laser fields. By using quantum-mechanical density matrix formalism, we obtain the equations of motion that govern the optical response of graphene in strong magnetic and optical fields. We found that by properly choosing the parameters of the system, the bistable behaviors and group velocity can be controlled. These results may have potential applications in telecommunication and optical information processing.

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1 © 2015 Astro Ltd Printed in the UK
1. Introduction
In the past few years, graphene has drawn considerable
interest because of its potential applications in nanoelec-
tronics and condensed matter physics. It is shown that its
magneto-optical properties and peculiar thin graphite layers
lead to multiple absorption peaks and unique selection rules
in graphene [1–4]. Recent progress in growing high-quality
epitaxial graphene has drawn intense interest since the com-
bination of unusual electronic properties and excellent opti-
cal properties, provide a way for new applications in infrared
optics and photonics [5–7]. The nonlinear electromagnetic
response of classical charges in magnetized graphene with
linear energy dispersion has been theoretically studied [8].
Moreover, the linear and nonlinear optical response of gra-
phene, nonlinear frequency conversion of terahertz (THz)
surface plasmons as well as the generation of polarization-
entangled photons based on nonlinear optical interaction in
strong magnetic elds have been discussed [9–12]. Bilayer
graphene has potential applications in new compact photonic
and optoelectronic devices due to its giant and tunable sec-
ond-order optical nonlinearity [13]. Wu and et al have theo-
retically studied the formation and ultraslow propagation
of infrared solitons in graphene under an external magnetic
eld [14]. They found that by properly choosing the param-
eters of the system, the formation and ultraslow propaga-
tion of infrared spatial solitons originate from the balance
between nonlinear effects and the dispersion properties of
the graphene under infrared excitation. Therefore, graphene
has become important and interesting from the practical
viewpoint of quantum physics. Many interesting physical
phenomena, such as electromagnetically induced transpar-
ency [15–17], four-wave mixing [18, 19], solitons [20–23],
optical bistability [25, 26] and so on [27–31] have arisen
from interaction of optical elds with nonlinear media. In
particular, optical bistability has drawn considerable inter-
est due its potential applications in the future photonic
devices. Based on atomic coherence and quantum interfer-
ence, many proposals have been proposed for optical bista-
bility and multistability in atomic systems [32–35], doped
Laser Physics Letters
Slow light propagation and bistable
switching in a graphene under an external
magnetic eld
SeyyedHosseinAsadpour
1
, HamidRezaHamedi
2
and
HamidRahimpourSoleimani
1
1
Department of Physics, University of Guilan, Rasht, Iran
2
Institute of Theoretical Physics and Astronomy, Vilnius University, A. Gostauto 12, LT-01108 Vilnius,
Lithuania
E-mail: S.Hosein.Asadpour@gmail.com
Received 25 September 2014, revised 8 February 2015
Accepted for publication 12 February 2015
Published 6 March 2015
Abstract
In this letter, we show the possibility of controlling the optical bistability and group index
switching in graphene under the action of strong magnetic and infrared laser elds. By using
quantum-mechanical density matrix formalism, we obtain the equationsof motion that govern
the optical response of graphene in strong magnetic and optical elds. We found that by
properly choosing the parameters of the system, the bistable behaviors and group velocity can
be controlled. These results may have potential applications in telecommunication and optical
information processing.
Keywords: group velocity, optical bistability, graphene
(Some gures may appear in colour only in the online journal)
S H Asadpour et al
Printed in the UK
045202
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Laser Phys. Lett. 12 (2015) 045202 (6pp)

S H Asadpour et al
2
solid states materials [36–38], semiconductor quantum wells
and quantum dots [39–45].To the best of our knowledge, the
OB and OM properties of a weak probe light under slow
or fast light propagation in graphene have not been studied
by any research groups to date. Recently, optical solitons
have also been investigated in graphene both experimentally
and theoretically [46–49]. The direct evidence of solitons in
graphene by means of molecular dynamics simulations and
mathematical analysis has been demonstrated [50]. It has
been reported that graphene exhibits an extremely strong
nonlinear optical response in the THz or infrared regime
[51–54]. Therefore, graphene structures have many potential
applications in the realm of nonlinear optics, although these
have not been explored yet. In this letter, according to the
quantum optics and solid-material scientic principles, we
study the behaviors of optical bistability and multistability
under slow or fast light propagation in the infrared region in
graphene under an external magnetic eld. Our investigation
of light propagation and bistable behaviors of a probe eld in
graphene may have potential applications in quantum infor-
mation processing.
2. Model and equations
A doped graphene system in the presence of a strong magnetic
eld with four level energy levels is considered in lambda con-
guration (gure 1). According to the peculiar selection rules
of graphene, i.e.
Δ =±n 1
(n is the energy quantum number)
as opposed to
Δ =±n 1
for electrons, the chosen transitions
between Landau levels are dipole allowed. Such a system has
been already used for studying the giant optical nonlinear-
ity, nonlinear frequency conversion, generation of entangled
photons and formation of ultraslow solitons [10–14]. Optical
transitions between the adjacent landau levels (LLs) in gra-
phene fall into the infrared to THz region for a magnetic
eld in the range 0.01–10 T:
ωℏ ≃
B36 (Tesla)meV
.
c
The
bichromatic electric elds consisting of the pulsed probe
eld and the continuous-wave (CW) control eld can be
written as
⃗
⃗
ω
⎯→
=−+
⎯→
+
E eE tk
rc
cexp( ii.)
..
11111
and
=
⃗
⃗
Ee
E
23 22
ωω−+ +−++
⃗
⃗⃗
⃗
⃗
tkreEtkr
cc
exp( ii.) exp( ii.)
..
22 33 33
Here
⃗
e
j
and
⎯→
k
j
are related unit vectors of the polarization eld
and the wave vector with the slowly varying envelope
=E j(1,2,3).
j
It is assumed that the probe eld with right-
hand circular polarization and corresponding amplitude and
carrier frequency E
1
and ω
1
interacts with the inter-Landau
level transition
↔14.
The linearly polarized CW control
eld
⎯→
E
23
with carrier frequency ω
2
(ω
2
= ω
3
) is applied to
drive the intra-Landau transition
↔23
via the σ
+
compo-
nent and
↔34
via the σ
−
component. The unit polariza-
tion vectors
⃗
e
2
and
⃗
e
3
can be expressed as
⃗
=+
̂
̂
exy(i)/ 2
2
and
⃗
=+
̂
̂
exy(i)/ 2
3
in the x–y plane of the graphene. In
the absence of an external optical eld, the effective mass
Hamiltonian for a single layer graphene (in the x–y plane)
under the magnetic eld
̂
Bz
(perpendicular to the plane of
graphene) can be given by:
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
ππ
ππ
ππ
ππ
=
−
+
+
−
̂̂
̂̂
̂̂
̂̂
̂
Hv
0i
00
i
000
000 i
00
i0
F
xy
xy
xy
xy
0
(1)
where
γ=ℏ≈
−
va3/(2 )10ms
F
0
61
(
γ ∼ 2.8eV
0
and
=aA1.42
o
are the nearest-neighbor hopping energy and C–C
spacing) is a band parameter (Fermi velocity), π
⎯→
=
⎯→
+
⎯→
̂
̂
peAc
/
denotes the generalized momentum operator,
⎯→
̂
p is the elec-
tron momentum operator, e is the electron charge, and
⎯→
A
is the
vector potential, which is equal to (0, Bx) for a static magnetic
eld. In order to consider the interaction with incident optical
eld, we need to add the vector potential of the optical eld
ω
⎯→
=
⎯→⎯→
=
⎯→
+
⎯→
AcEE
EE(
i/an
d)
opt
12
3
to the vector potential of the
magnetic eld in the generalized momentum operator
π
⎯→
̂
in
the Hamiltonian. The resulting interaction Hamiltonian can be
expressed in the following form:

⃗
⃗
σ=Hv
e
c
A
..
Fint opt
(2)
A standard time-evolution equation for the density matrix
of Dirac electrons in graphene coupled to the infrared
laser elds can be obtained by utilizing Liouville’s equa-
tion
ρ
ρρ
∂
∂
=−
ℏ
+
̂
̂̂
̂
̂
t
HR
i
[,
](
).
int
Here
ρ
̂
̂
R
()
indicates incoher-
ent relaxation which may originate from disorder, interaction
with phonons, and carrier–carrier interactions. The density
matrix equations of motion for the coupled system can be
written as follows:
Figure 1. (a) LLs near the K point superimposed on the electronic
energy dispersion without a magnetic eld
=±E vp.
F
The
magnetic eld condenses the original states in the Dirac cone
into discrete energies. The LLs in graphene are unequally spaced:
∝
B . (b) Energy level diagram and optical transitions in graphene
interacting with two continuous-wave control elds 2 and 3 and a
weak pulsed probe eld 1. The states
1, 2, 3and 4
correspond
to the LLs with energy quantum numbers n =−2,−1, 0, 1,
respectively. Graphene monolayer is a one-atom-thick monolayer of
carbon atoms arranged in a hexagonal lattice, which we will treat as
a perfect (2D) crystal structure in the x–y plane.
Laser Phys. Lett. 12 (2015) 045202

S H Asadpour et al
3
ρΩρΩρ
ργρΩρΩρ
ργρΩρΩρΩρΩρ
ργρΩρΩρΩρΩρ
ρ
γ
ρΩρΩρ
ρΔω
γ
ρΩρΩρΩρ
ρΔω
γ
ρΩρρ Ωρ
ρΔω
γγ
ρΩρρ Ωρ
ρΔω
γγ
ρΩρΩ
ρΩ
ρ
ρΔωΔω
γγ
ρΩρ
Ωρ ρΩρ
=−
=− +−
=− ++−−
=− ++ −−
=− +−
=− +++−
=− ++ −+
=− +
+
+−+
=− +
+
++ −
=− −+
+
+
+−−
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
˙
ii
˙
ii,
˙
iiii,
˙
iiii,
˙
2
ii,
˙
i
2
ii
i,
˙
i
2
i( )i ,
˙
i
2
i( )i ,
˙
i
2
ii
i,
˙
ii
2
i
i( )i
11
1
41
1
14
22 222
2
32
2
23
33 333
2
23
3
43
2
32
3
34
44 444
1
14
3
34
1
41
3
43
21
2
21
2
31
1
24
31
2
3
31
2
21
3
43
1
34
41
1
4
41
1
11 44
3
31
32
2
32
32
2
22 33
3
42
42
1
42
42
1
12
3
32
2
43
43
12
43
43
1
13
3
33 44
2
42
(3)
where
Δωε εωεε ω=− ℏ− =−ℏ−
==−==
()/( )/
nn nn112110
3
and
Δωε εω=− ℏ−
==−
()/
nn20
12
represent the corresponding
frequency detunings, and ε ω
=ℏnn
sgn( )
nc
is the energy
of the Landau level for electrons near the Dirac point, with
ω=±±=
n
vl
0, 1, 2, ...,
2/
,
cFc
and
=ℏlc
eB/
c
implies the
magnetic length.
Ωμ Ωμ=ℏ=ℏ
⃗
⃗
⃗
⃗
eE eE(.)/(2 ), (.)/(2 ),
1
41
11 2
32
22
and
⃗
Ωμ
=
⎯→
ℏ
eE(.)/(2
)
3
43
33
are the corresponding one-
half Rabi-frequencies with
μμ==
⃗ ⃗
⃗
mn em
rn
.
mn
εε σ= ℏ−
⃗
emvni/
nm F
being the dipole matrix element for
the relevant optical transition.
γ =j(2,3,4),
j
corresponds to
the decay rate of the state
j .
The set of equations(3) can be
solved numerically to obtain the steady-state response of the
medium. In fact, the response of the medium to the applied
elds is determined by the susceptibility χ, which is dened as:
⃗
χ
μ
ε
ρ=
N
E
2.
.
r
14
1
41
(4)
Here N and ε
r
are the sheet electron density of graphene and
the substrate dielectric constant, respectively.
Now, we consider a medium of length L composed of the
above described graphene system immersed in a unidirec-
tional ring cavity [25]. For simplicity, we assume that both
mirrors 3 and 4 are perfect reectors, and the reection and
transmission coefcients of mirrors 1 and 2 are R and T (with
R + T = 1), respectively.
Under slowly varying envelop approximation, the dynam-
ics response of the probe beam is governed by Maxwell’s
equations:
ω
ε
ω
∂
∂
+
∂
∂
=
E
t
v
E
z
P
i
2
()
F
r
111
1
(5)
where P(ω
1
) is the induced polarization. For a perfectly
tuned cavity, the boundary conditions in the steady-state limit
between the incident eld
E
I
1
and transmitted eld
E
T
1
are:
=E L
E
T
()
,
T
1
1
(6a)
=+
E TE RE L(0
)(
),
I
1
1
1
(6b)
where L is the length of the atomic sample. Note that the
second term on the right-hand side of equation (6b) is the
feedback mechanism due to the reection from mirrors. It
is responsible for the bistable behavior, so we do not expect
any bistability when R = 0 in equation(6b). According to the
mean-eld limit and by using the boundary condition, the
steady-state behavior of an elliptically polarized transmitted
eld is given by:
ρ=−yxC2i
41
(7)
where
μ=ℏ
yET/
I
14
1
and
μ=ℏ
xET/
T
14
1
are the normal-
ized input and output eld, respectively. The parameter
ωμ ε=ℏC
NL
vT
/2
rF1
14
2
is the cooperative parameter for atoms
in a ring cavity. The transmitted eld depends on the incident
probe eld and the coherence terms ρ
41
via equation(7).
3. Results and discussion
In this section, the behaviors of cavity input–output eld
and group index of weak probe light are discussed in the
assumption of specic parameters. We assume that the Rabi
frequency Ω
1
of the pulsed probe eld is much smaller than
that of the control elds Ω
2
and Ω
3
, so that the Dirac electrons
are initially populated in their ground state
1
without deple-
tion. As a result,
ρ ≃ 1
11
whereas
ρ ρρ≃≃≃ 0.
22 33 44
That is
to say, we assume that state
1
is fully occupied while states
2, 3and 4
are empty; i.e. the Fermi level is between LLs n
=−2 and n =−1. It is noted that the carrier frequency of the probe
eld can be estimated approximately the same amount of the
transition frequency
ωε
εω=− ℏ= +
==−
()/
(
21
)
,
nn c41 12
which is on the order of
ω ∼×
−
1.86 10 s
41
14 1
for graphene
at the magnetic eld of the value B = 1 T [11–14]. When the
magnetic eld reaches up to 5 T, the transition frequency is
estimated to be
ω ∼×
−
3.87 10
s.
41
14 1
In the present work, we
take the magnetic eld B = 3 T for example, at this moment
Figure 2. Output led versus input eld for different values of
frequency detuning of coupling laser. The selected parameters are
γ γγγγΩΩγ=× ====
−
310s , 0.05 ,,
,
3
13 1
2343
23
3
and
Δωγ= 3.5 .
1
3
0 5 10 15 20 25 30 35 40
0
5
10
15
20
25
30
35
Input field |y|
Outputfield |x|
∆ω
2
=5
∆ω
2
=0
Laser Phys. Lett. 12 (2015) 045202

S H Asadpour et al
4
ω ∼
−
10
s.
c
14 1
According to the numerical estimate based on
[9, 14], we can take a reasonable value for the decay rate
γ =×
−
310s ,
3
13 1
and assuming
γ γγ γ==,0.05,
4323
these val-
ues depend on the sample quality and the substrate used in
experiments [11–14]. The dipole moment between the transi-
tions
↔14
in the graphene has a magnitude of the order
μεε
⎯→
∼ℏ −∝ev B/( )1
/.
F
14
41
The electron concentration
is
≃×
−
N 510cm
12 2
and the substrate dielectric constant is
ε ≃ 4.5
r
[55]. In gure2 we show the effect of the frequency
detuning of the control eld on the behavior of the cavity
input–output eld. It can be found that the switching between
OM and OB can be obtained by the frequency detuning of the
coupling eld. It will be interesting to see from gure2 that
the OM behavior disappears when the detuning of the cou-
pling eld is zero. That is to say that, the multistable to bistable
behavior can be controlled very easily by tuning the detuning
of the coupling eld. The main reason for the above phenom-
ena is that the changed frequency detuning of the coupling
eld will modify the absorption and the Kerr nonlinearity of
the medium, which creates the multistable behavior disap-
pears. In gure3, we plot the input–output eld curves for
different values of Ω
2
. It is shown that the threshold of OB is
reduced by increasing the intensity of the coupling eld. The
reason for the above result can be qualitatively explained as
follows. In fact, the states
1, 3and 4
are comprised of a
usual three-level lambda type conguration which owns the
property of the electromagnetically induced transparency. By
applying an increasingly strong coupling eld between states
2and 3,
the absorption of the weak probe eld on the tran-
sition
4to1
is increased, which makes the hysteresis cycle
change dramatically. In order to gain deeper insight into the
above phenomenon, we plot a graph of absorption versus
the probe eld detuning in gure3(b). In the case of Ω
2
= 1,
the medium becomes transparent for the probe eld at reso-
nance and splits into two peaks when the control laser eld
is enhanced. A strong absorption peak occurs at resonance
and with increasing of the intensity of the coupling eld, the
height of the central peak increases gradually. In gure4, we
plot the absorption (a) and dispersion (b) versus coupling
Figure 3. (a) Output eld versus input eld and (b) absorption spectrum versus probe eld detuning for different values of coupling laser
eld. The selected parameter
Ωγ= 10
3
3
and others are the same as gure2.
0 5 10 15 20 25 30 35
0
2
4
6
8
10
12
14
16
18
20
Input field |y|
(a)
−10 −5 0 5 10
0
1
2
3
4
5
6
7
x 10
−3
∆
P
Absorption
(b)
Ω
2
=1
Ω
2
=10
Ω
2
=15
Output field |x|
Figure 4. Absorption (a) and dispersion (b) of weak probe light versus coupling laser eld Ω
2
. The selected parameters are the same as
gure3.
0 5 10 15
0
1
2
3
4
5
6
7
x 10
−3
Ω
2
Absorption
(a)
0 5 10 15
−1
0
1
2
3
4
5
x 10
−5
Ω
2
Dispersion
(b)
Laser Phys. Lett. 12 (2015) 045202

S H Asadpour et al
5
laser eld (Ω
2
). We nd that by enhancing the parameter Ω
2
,
the absorption increases and dispersion switches from nega-
tive to positive values. Therefore, the group velocity of the
light pulse can be controlled from superluminal to subluminal.
As is well known, the imaginary part and real part of suscep-
tibility determine the absorption and dispersion of the system,
respectively. The group velocity index is dened by
ωω
ω
ω
== +
n
c
v
n
n
() ()
d
d
g
g
11
1
1
(8)
where n(ω
1
) is the refractive index and
ωχ=+n
() 1Re[
].
1
When
−≻n 10,
g
the group velocity is smaller than the
velocity of light in vacuum, which means slow light can be
obtained. Conversely, when
−≺n 10,
g
the group velocity is
larger than the velocity of light in vacuum, and then fast light
can be obtained. From gure5, it is shown that the switch-
ing from superluminal to subluminal light propagation occurs
for an appropriate value of the strength of the coupling eld.
It is obvious that the coupling eld strength can be consid-
ered as a knob for changing the probe light propagation from
superluminal to subluminal speed. In gure6, we display the
dependence of OB on the cooperation parameter C. We nd
that the cooperation parameter C is directly proportional to
the number density N of the graphene center sample, and an
increase in the number density will enhance the absorption of
the sample, which accounts for the raise of the bistable thresh-
old when the cooperation becomes larger.
4. Conclusion
In conclusion, we have proposed a new scheme for realizing
OB and OM in a graphene system under an external mag-
netic eld inside an optical ring cavity. We nd that due to
the remarkable electronic properties and selection rules near
the Dirac points, by properly choosing the parameters of the
system, the bistable behaviors and group velocity can be con-
trolled. These results may have potential applications in tele-
communication and optical information processing.
Acknowledgment
This work is supported by the Iran Nanotechnology
Initiative Council. H R Hamedi gratefully acknowl-
edges the support of Lithuanian Research Council (No.
VP1-3.1-ŠMM-01-V-03-001).
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Figure 5. Group index versus probe eld detuning. Solid line
corresponds to
Ωγ= ,
2
3
dashed line corresponds to
Ωγ= 10
2
3
and
dotted line corresponds to
Ωγ= 15 .
2
3
The other parameters are the
same as gure3.
−10 −5 0 5 10
−0.5
0
0.5
1
1.5
2
2.5
∆
P
Group Index
Ω
2
=γ
3
Ω
2
=10γ
3
Ω
2
=15γ
3
Figure 6. Output eld versus input eld for different values
of the cooperation parameter C. The selected parameters are
ΩγΩγΔω γΔω== ==15 ,10, 3.5 ,0
2
3
3
3
1
3
2
and others are the
same as gure3.
0 5 10 15 20 25 30 35 40 45
0
5
10
15
20
25
30
Input field |y|
Outputfield |x|
C=150
C=175
C=200
Laser Phys. Lett. 12 (2015) 045202

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