Temporal evolutional absorption behaviors of graphene under Landau quantization

Author’s Accepted Manuscript Temporal evolutional absorption behaviors of graphene under Landau quantization H.R. Hamedi, M. Sahrai

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To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 3 March 2016 Revised date: 9 May 2016 Accepted date: 9 October 2016 Cite this article as: H.R. Hamedi and M. Sahrai, Temporal evolutional absorption behaviors of graphene under Landau quantization, Physica E: Low-dimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2016.10.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Temporal evolutional absorption behaviors of graphene under Landau quantization H. R. Hamedi1,*, M. Sahrai2 1

Institute of Theoretical Physics and Astronomy, Vilnius University, Saulėtekio 3, Vilnius LT-10222

Lithuania 2

Research Institute for Applied Physics, University of Tabriz, Tabriz, Iran

[email protected] [email protected] *

Corresponding author. Tel.: +37067218909

Abstract We investigate the evolutional absorption behaviors of Landau- quantized graphene structure based on the transient solution to the density matrix equations of the motion. The impact of various system parameters on temporal evolution of probe absorption is studied. In addition, the required times for switching the high- absorption case to the zero- absorption (transparency) of a probe field is discussed. Due to unusual optical and electronic characteristics of graphene resulting from linear, massless dispersion of electrons near the Dirac point and the chiral character of electron states, our study may have potential applications in telecommunication, biomedicine, and optical information processing and may cause significant impact on technological applications. Keywords: Transient evolution; Switching time; Graphene.

PACS number(s): 42.50.Gy; 74.25.Gz

1. Introduction In the past few decades, comprehensive research in the area of quantum and nonlinear optics has led to considerable attention in the study of optical response of the atomic system interacting with a number of coherent fields. Due to many limitations for practical applications in two- level traditional systems, researchers are more interested in multi-level configurations. An additional field probing another excitation path can lead to an induced coherence to the system. The induced coherence can lead to various quantum optical effects 1

like electromagnetically induced transparency (EIT) [1, 2], slow light [3- 5], lasing without inversion [6, 7], optical soliton [8- 10], giant Kerr nonlinearity [11- 13], multi-wave mixing [14–17], and so on [18- 24]. EIT, as one of the most interesting consequences of preparing an atomic system in a multilevel configuration, plays a crucial role to control the optical response and related absorption of weak laser light. EIT phenomenon has been found important applications in quantum information science, such as the photon information storing and releasing in an atomic assemble [25], correlated photon pairs generation [26] and even the entanglement of remote atomic assembles [27], which may form the building blocks of the quantum communication and the quantum computation. In view of many proposals, the transient evolutional properties of the weak probe field via quantum interference such as transient-absorption, transientdispersion, and transient-gain without inversion are extensively studied in atomic and solid systems [28- 38]. It is also worthwhile to probe the transient behaviors in one-atom- thick materials such as graphene. As a single-atom-thick allotrope of C with abnormal two-dimensional (2D) Diraclike electronic excitations [39-41], various relativistic quantum phenomena can be generated in graphene which cannot be realized in general materials due to very unique optical transition selection rules in graphene [42-44]. Dirac’s equation can govern the transport of massless Dirac quasiparticle electrons in graphene with a linear energy-momentum dispersion relation [45- 47]. Dirac electrons can be adjusted by means of external electromagnetic fields [48]. Graphene has a large nonlinear response in its optical region of spectrum [49- 51]. Several nonlinear optical properties of graphene have been investigated under the action of a strong magnetic field such as four-wave mixing (FWM) and multi-wave mixing (MWM) [52], generation of entangled photons [53], giant optical nonlinearity [54- 59], formation of optical solitons [60, 61]. Recently, the formation and propagation of infrared solitons resulting from the balance between nonlinear effects and the dispersion properties of the graphene under infrared excitation have been reported [60, 61]. Based on these outstanding works, one of the present authors has studied numerically the slow light propagation [62], optical bistability [63], and steady-state absorption- dispersion properties [64] in graphene under an external magnetic field. In the current study, we aim to explore the temporal behavior of probe absorption in Landau- quantized graphene. We investigate how the system parameters can affect the transient response of the probe field. We also demonstrate that this system can be employed 2

as an optical switch in which a high- absorption case can be converted to the zero- absorption (transparency). Stemming from unusual electronic and optical properties of graphene stemming from linear, massless dispersion of electrons near the Dirac point and the chiral character of electron states, our proposed scheme may provide a route to explore optically controllable switches with application in optical information processing. 2. Model and Equations We consider a 2D graphene crystal with four energy levels in the presence of a strong magnetic field which forms a Ladder- type configuration. Due to attendance of a magnetic field, the linear dispersion relation of graphene causes unequally spaced Landau levels (LLs) with transition energies proportional to

B [54, 59]. It can be determined by the exotic

selection rules of graphene (  n  1 with n being the energy quantum number) that the selected transitions are dipole allowed. Under the condition of  n  1 and  n  1 , the right-hand circularly (RHC) and the left-hand circularly (LHC) polarized light beam are absorbed respectively [65]. Optical transitions between the adjacent LLs in graphene fall into to the infrared to THz domain for a magnetic field in the range 0.01-10T:

c  36 B(Tesla) meV .One can write the electric fields as E  E p  E23 ,

(1) with

E p  ep E p exp(i pt  ik p . r )  c. c. (2a)

E23  e2 E2 exp(i2t  ik2 . r )  e3 E3 exp(i3t  ik3 . r )  c. c. (2b) where, e j and k j are unit vectors of the polarization field and the wave vector with the slowly varying envelope E j ( j  p, 2,3) , respectively. The linearly polarized CW control field E23 mediates the intra-Landau transition 2  3 via the   component and 3  4 via the   component. In the x-y plane of the graphene, the unit polarization vectors e2 and e3 can be

3

expressed as e2  ( xˆ  iyˆ ) / 2 and e3  ( xˆ  iyˆ ) / 2 . A weak tunable right-hand circular polarization incident probe pulse with the amplitude E p and carrier frequency  p drives the intra-LL transition 1  2 . Probe transition 1  2

is also coupled with a weak

incoherent pumping field r . In presence of a magnetic field perpendicular to a single-layer graphene (in the x - y plane), the quantum interference and quantum coherence effects appear due to the optical pathways driven by the probe and control fields. The total Hamiltonian of this system under consideration (Fig. 1(b)) in the interaction picture and under the rotating wave and electric dipole approximations then reads (see Appendix I) I Hˆ int  ( p 2 1  2 3 2  3 4 3  h.c.)

(3)

 p 2 2  ( p   2 ) 3 3  ( p   2  3 ) 4 4 , with

the

corresponding

detunings

 p  ( n1   n2 ) /   p ,

are

2  ( n0   n2 ) /   p  2 , 3  ( n1   n2 ) /   p  22 . Here,  n  sgn(n) c n denotes

the

energy

of

the

n  0,  1,  2,..., c  2vF / lc ,

corresponding

one-

LL

where

half

for

electrons

lc 

c / eB

Rabi-

near is

the

frequencies

the

Dirac

point,

with

magnetic

length.

The

are

defined

as

 p  (21.ep ) E p / (2 ), 2  ( 32 .e2 ) E2 / (2 ) , and 3  (43 .e3 ) E3 / (2 ), with the dipole matrix

elements

for

mn  m  n  e. m r n 

the

relevant

optical

transition

being

i e m vF  n . n  m

Taking the advantage of Liouville’s equation

ˆ i ˆ {ˆ},   [ Hˆ intI , ˆ ]   t (4) the resulting density matrix equations of motion of Dirac electrons in graphene coupled to the infrared laser fields can be obtained as

4

 21  (i p 

2 2

 2 R)  21  i p ( 11   22 )  i 2 31 ,

31  (i ( p   2 ) 

3 2

 R) 31  i 2  21  i3  41  i p 32 ,

 41  (i ( p   2   3 )  32  (i 2 

3 2 2

 42  (i ( 2   3 )   43  (i3 

4 3 2

4 2

 R)  41  i3 31  i p  42 ,

 R) 32  i 2 (  22  33 )  i3  42  i p 31 ,

4 2 2

 R)  42  i3 32  i 2  43  i p  41 ,

)  43  i3 ( 33   44 )  i 2  42 ,

11   2  22  i p ( 12   21 )  2 R( 11   22 ), 22   2 22   3 33   4  44  i2 (  23  32 )  i p ( 12   21 )  2 R( 11   22 ), 33   3 33  i2 (  23  32 )  i3 (  43  34 ), 44   4 44  i3 ( 34   43 ). (5) ˆ {ˆ } shows incoherent relaxation which may originate from disorder, Note that in Eq. 5, 

interaction with phonons, and carrier-carrier interactions. Moreover, in Eq. 5, the decay rate of the state j is represented by  j ( j  2,3, 4) . The probe complex dielectric susceptibility χ of the system can be written as

   R  i i 

2 N .21  ,  r E p 21

(6) with N  5 1012 cm2 and  r  4.5 being the sheet electron concentration of graphene and the substrate dielectric constant, respectively [66- 69]. The real  R and imaginary  i parts of probe susceptibility given in Eq. 6 represent the dispersion and absorption of probe field, respectively. 4. Results and discussion Assuming  p

2 , 3 , the Dirac electrons are considered initially populated in their ground

state 1 without depletion. As a result, 11  1 , while other elements are zero. This indicates

5

the state 1 is fully occupied while other states are empty; i.e., the Fermi level is between LLs n  2 and n  1 . Under this assumption and utilizing the weak-field approximation [11], the steady-state solution for the coherence term  21 reads:

21  i p

w1w2  32 , w1w2 w3  w222  w332

(7)

where w 1  i ( p   2 ) 

3 2

 r , w2  i( p   2  3 ) 

4 2

 r , w3  i p 

2 2

 2r .

Equation 7 gives the dependence of absorption- dispersion properties of the probe field to the system parameters, i.e., intensity and frequency detuning of the driving fields. First, we intend to discuss the steady- state properties of weak probe field in this system through the analytical solution given in Eq. 7. To this end, in our simulations we require to take reasonable values for the decay rates. The carrier frequency of the probe field can be approximated by an amount of the transition frequency 21  ( n1   n2 ) /  ( 2  1) c , which is on the order of 21  2.39 1014 s 1 for graphene at the magnetic field of the value B  1T [49, 53]. Intensifying the magnetic field up to 5T, the transition frequency can be

approximated to be 21 =5.34 1013 s 1 . We assume here, for example, B=3T, at this moment the transition frequency will be approximately c  1014 s 1 . The reasonable value

 3  3 1013 s 1 can be taken for the decay rate as stated in the numerical estimate based Refs [54, 65]. In addition, in our simulations we set  2   3   4 . These values depend on the sample quality and the substrate used in experiments [49, 52-54]. The magnitude of dipole moment

between

the

transition

1  2 in

the

graphene

is

on

the

order

12  evF / ( n1   n2 )  1/ B [65- 68]. Upon these conditions, the typical probe susceptibility of the system against probe field detuning  p and in resonance condition 2  3  0 is shown in Fig. 2. As can be seen from Fig. 2(a), when 2  3  3 3 , three absorption peaks appear in the medium which is due to the interacting double dark resonances. The slope of linear dispersion is negative around zero probe detuning which corresponds to superluminal light propagation. When neglecting the 6

second control field 3 (i.e., 3  0 but 2  3 3 ), it is observed from the susceptibility profile (Fig. 2(b)) that only two absorption peaks emerge at both sides of zero probe detuning, indicating a very weak absorption at line center due to the effect of control field

2 . The above result can be easily understood through the dressed state representation. Generally, the dressed eigenstates due to two control fields 2 , 3 are [19]

L 

K [2 2  3 4 ] 2

1 3, 2

(8a)

L0  K[3 2  2 4 ],

Where K 

1 22  32

(8b)

. The corresponding eigenenergies are E   

K

, and E 0  0 .

When 2  3   , we obtain

L 

L0 

1 [( 2  4 ) 2

3 ],

(8a)

1 [ 2  4 ], 2

(8b)

With the eigenenergies being E    2 , and E 0  0 . As a result, the origin of three absorption peaks bserved in Fig. 2(a) can be realized by Eq. 8. When 3  0 , we have

L 

1 (2 2

3 ),

L0  4 ,

(9a)

(9b)

Which indicates that in this condition, L 0

coincides with the bare state 4 , and thus is

decoupled from the fields. Thus, two absorption peaks appear which are separated by the energy splitting

7

2 2 .

To be of practical interest, in the following, we consider the propagation of a weak Gaussianshaped probe pulse

 (0, )  0p e[(  )/ ] , 2

(10)

0

Where here  0 is the temporal width of the input pulse, and  is related to the location of peaks. To this end, we need to solve the set of equations 5 together with the Maxwell’s wave equation along the propagating direction zˆ

 p ( z, t ) z



where  

1  p ( z, t )  i 3 21 ( z, t ), vF t N 1 21.e p 2 r v F  3

(11)

2

is the coupling constant. The solution of Maxwell- Bloch equations

11 and 5 provides the complete evolution of the probe pulse in the medium. For simplicity, we take the Rabi frequency of the weak probe pulsed field as  p (z , t )  0p (z , t ) , where  0p is a real constant representing the maximal values of Rabi frequency before the pulse

enters the medium (i.e., at z  0 ), and  ( z, t ) is the dimensionless spatiotemporal pulseshaped function. Moreover, we select   30 /  3 and  0  180 /  3 . In Fig. 3, the threedimensional (3D) plots of pulse propagation for the same set of values as Fig. 2 are plotted. It can be observed from Fig. 3(a) that when 2  3  3 3 , the weak probe pulse confronts broadening and it is absorbed by the medium after almost a short propagation distance. However, as shown in Fig. 3(b), for 2  3 3 and 3  0 , the weak pulse is transmitted through the sample nearly without any considerable losses and broadening while it can still preserve its shape for quite a long propagation distance. This result is in a good agreement with the outcome given in Fig. 2. In what follows, we discuss the effect of system parameters, i.e. 2 , 3 as well as  2 , 3 ,  p on transient evolution of the probe absorption. We display the time evolution of absorption in graphene system for different values of  2 , 3 ,  p in Figs. 4(a)- 4(c), respectively. It can be seen from the figures that under the resonance condition, probe absorption never manifests periodic gain-absorption; the absorption exhibits an oscillatory behavior for quite a short time and finally reaches a positive steady state value which corresponds to the probe absorption. However, by increasing each of parameters  2 , 3 , and  p as shown in Figs. 4(a)- 4(c), the 8

steady- state absorption decreases gradually until it attains a very small value. In this case, the graphene sample becomes nearly transparent to the probe field. Next, the impact of the control fields intensities 2 , 3 on transient and steady state absorption spectra is explored in Fig. 5 and 6, respectively. It is observed from Figs. 5(a) and 6(a) that the absorption coefficients are very sensitive to  2 and 3 . As expected, the absorption finds the steady- state after passing intense oscillatory behaviors. However, we realize that the steady- state curves behave differently for  2 and 3 . Specifically, increasing

 2 leads to a significant decreasing of steady-state probe absorption, while the steady- state absorption enhances by increasing 3 . In order to better understanding of this behavior, we plot in panel (b) of both Figs. 5 and 6 the steady- state probe absorption against  2 and 3 . It is clearly seen that the above comment is held; the probe absorption decreases (increases) by further increasing  2 ( 3 ). In order to demonstrate the controllability of temporal evolution of probe absorption through r parameter, we plot in Fig. 7 the time-dependent absorption profile of the probe field. It is

found that by increasing r , both the oscillatory amplitudes and the steady-state values of absorption reduces dramatically, so that for quite large values of r (for instance r  5 3 ), the transparency becomes dominant in the graphene medium (Fig. 7(a)). As a matter of fact, applying an increasingly incoherent pumping rate to the probe transition 1  2 reduces significantly the amount of steady- state probe absorption at zero probe detuning as shown in Fig. 7(b). Physically, for small values of incoherent pumping r , 11

22 , while large

values of r can moderate the population distribution between levels 1 and 2 . In this case, the population difference between levels 1 and 2 reduces which is because of this point that an increasingly incoherent pumping rate pumps more population to the upper level 2 , while less population remains in level 1 so that 11  22

0 . To have an estimate of the

population distribution contributed in probe absorption, the population difference distribution

11  22 of the graphene system is plotted in Fig. 7(c). It can be seen from Fig. 7(c) that after a very rapid oscillation, the steady-state values of population difference decreases by increasing r . This result supports our previous comments given in Figs. 7(a) and 7(b).

9

We now demonstrate that this system can be employed as an optical switch between high absorption and nearly transparency (negligible absorption), through modulating the intensity of control fields as well as the rate of incoherent pumping. Indeed, we are looking for the required switching time for changing the high absorption case to the nearly transparency or vice versa by proper manipulating the  2 (Fig. 8(a)), 3 (Fig. 8(b)), and r (Fig. 8(c)). Figure 8(a) shows that under the effect of  2 the required time for switching from high- absorption to nearly transparency or vice versa is about 15 /  3 (Fig. 8(a)), while under the effect of 3 it becomes 13 /  3 (Fig. 8(b)). More interestingly, under the action of incoherent pumping, the switching time form high- absorption to zero- absorption becomes 1/  3 , whereas we obtain

15 /  3 for the switching time from zero- absorption case to high- absorption one, as shown in Fig. 8(c). An optical switch, in which the propagation of a light pulse can be controlled with another pulse, has potential application in optical information processing and transmission. A high speed optical switch is an important technique for quantum information network and communication, and may be useful for understanding the switching feature of EIT-based systems.

5. Conclusions To summarize, we have studied the transient evolution of probe absorption in Landauquantized graphene by using density matrix approach. The effects of different parameters of system such as intensity and frequency detunings of driving fields on time- dependent as well as steady- state response of graphene structure are explored. We have also investigated the switching process in which high- absorption can be converted to the zero- absorption of a probe field. Because of unusual electronic and optical properties of graphene resulting from linear, massless dispersion of electrons near the Dirac point and the chiral character of electron states, our proposed scheme may provide a route to explore optically controllable switches with application in optical information processing.

Acknowledgments H. R. Hamedi gratefully acknowledges the support of Lithuanian Research Council (No. VP1-3.1-ŠMM-01-V-03-001). 10

Appendix I The effective mass Hamiltonian, in the absence of an external optical field for a single layer graphene under the action of magnetic field Bzˆ reads [70, 71]

 0 ˆ  x  iˆ y ˆ H 0  vF   0   0

ˆ x  iˆ y 0

0 0

0 0

0 ˆ x  iˆ y

   ˆ x  iˆ y   0  0 0

I(a)

where, vF  3 0 / (2 a)  106 m / s and ˆ  pˆ  eA / c indicate the Fermi velocity (band parameter) and the generalized momentum operator, respectively (  0  2.8eV is nearesto

neighbor hopping energy, a  1.42 A shows the C-C spacing, pˆ is the electron momentum operator, e denotes the electron charge, and A is the vector potential, which is equal to

(0, Bx) for a static magnetic field). The eigenenergies of discrete LLs for the magnetized graphene can be obtained by solving the effective mass Schrödinger equations Hˆ 0   . Generally, the Hamiltonian near the K point can be written as Hˆ 0  v F ˆ .ˆ , where

ˆ  (ˆ x , ˆ y ) is a Pauli matrices vector. The eigenfunction is then determined by two quantum numbers n (n  0, 1, 2,...) and the electron wave vector k y along the y direction [20, 54, 59],

 n ,k (r )  y

Cn L

e

 ik y y

 sgn(n)i n 1 n 1   , n   i n  

I(b)

where  1(n  0)  Cn   1 ,  2 (n  0) 

and

11

,I(c)

n 

H n (( x  lc2 k y ) / lc ) 2 n !  lc n

1 x lc k y 2  ( ) 2 lc 2

e

,

I(4)

with l c  c / eB being magnetic length and H n (x ) denotes the Hermite polynomial. The eigenenergy is  n  sgn(n ) c

n with c  2v F / l c

In fact, compared to LLs of a conventional 2D electron/hole system with a parabolic dispersion, LLs in graphene are unequally spaced and their transition energies are

B [59]. Merging the eigenenergy of the graphene with the selected energy

proportional to

levels in Fig. 1(b), one can simpobtainlify the Hamiltonian of system in absence of optical fields as

Hˆ 0  (1 1 1   2 2 2   3 3 3   4 4 4 ),

I(5)

In order to inspect the interaction with the incident optical fields, one needs to add the vector potential of the optical field ( Aopt  icE / , E  E p  E23 ) to the vector potential of the magnetic field in the generalized momentum operator ˆ in the Hamiltonian. The resulting interaction Hamiltonian can be expressed as

e Hˆ int  vF . Aopt , c

I(6)

Eq. (4) shows that unlike the case of an electron with a parabolic dispersion relation, there are no higher-order terms for the interaction Hamiltonian ( H int ) in graphene in the vicinity of the Dirac point. So, H int remains linear with respect to Aopt even for a relatively strong optical field. The interaction Hamiltonian does not include the momentum operator, it is only determined by the Pauli matrix vector and proportional to vector potential Aopt . By inserting the complete set of states {1 , 2 , 3 , 4 } in the generated interaction Hamiltonian, one can get

e  i t Hˆ int  I .vF . Aopt .I   21E p e p 2 1  32 E2e i2t 3 2  43 E3e i3t 4 3  h.c. c   ( p 2 1 e

 i p t

 2 3 2 e

 i2t

 3 4 3 e

 i3t

 h.c.)

where the complete set of states is I   i i (i  1, 2,3, 4) . i

12

I(7)

Then we can get the total Hamiltonian of the system, i.e., Hˆ  Hˆ 0  Hˆ int , where in order to simplify this relation, we assume that the state 1 is the zero potential reference. In the interaction picture, with the rotating wave approximation and the electric dipole approximation, the total Hamiltonian can be expressed as Eq. 3. Inserting Eq. 3 into Eq. 4, the set of density matrix equations 5 can be then easily obtained. References [1] S.E. Harris, Phys. Today 50 (1997) 36. [2] Y. Wu, X. Yang, Phys. Rev. A, 71 (2005) 053806. [3] L.V. Hau, S.E. Harris, Z. Dutton, C.H. Behroozi, Nature (London) 397, (1999) 594. [4] S. M. Ma, H. Xu, B.S. Ham, Optics Express 17 (2009)14902. [5] M. M. Kash, V.A. Sautenkov, A.S. Zibrov, L. Hollberg, G.R. Welch, M.D. Lukin, Y. Rostovtsev, E.S. Fry, M.O. Scully, Phys. Rev. Lett 82 (1999) 5229. [6] ZhipingWang, Benli Yu. Appl Phys A 109 (2012) 725–729. [7] S. E. Harris, Phys. Rev. Lett. 62 (1989) 1033 [8] Y. Wu, L. Deng, Phys. Rev. Lett. 93 (2004)143904. [9] Y. Wu, Phys. Rev. A 71 (2005)053820. [10] Wen-Xing Yang, Jing-Min Hou, and Ray-Kuang Lee, Phys. Rev. A 77 (2008) 033838 [11] Y. Niu, Sh. Gong, R. Li, Zh. Xu, and X. Liang, Optics Letters (2005) 30 ( 24) 3371-3373 [12] Xiang-an Yan, Li-qiang Wang, Bao-yin Yin, Wen-juan Jiang, Huai-bin Zheng, Jian-ping Song , Yan-peng Zhang, Physics Letters A 372 (2008) 6456. [13] Wen-juan Jiang, Xiang’an Yan, Jian-ping Song, Huai-bin Zheng, Chuanxing Wu, Baoyin Yin, Yanpeng Zhang. Optics Communications 282 (2009)101–105.

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Figure 1. (a) Landau levels near the Dirac point superimposed on the linear electron dispersion without the magnetic field E  vF p . (b) Energy level diagram and optical transitions in graphene interacting with two continuous-wave control fields and a weak pulsed probe field. The states 1 , 2 , 3 and 4 correspond to the LLs with energy quantum numbers n  2, 1,0,1 , respectively Figure 2: Plots of the probe susceptibility against  p . The selected parameters are

 3  3 1013 s 1 ,  2   4   3 , r  0,  p  0.01 3 , 3   2  0 , 2  3 3

(a)

3  3 3 ,

(b)

3  0. Figure 3: Three dimensional plot of probe field intensity with 0p  0.01 3 in the medium against time and distance for (a) 3  3 3 , (b) 3  0 and the other parameters are the same as Fig. 2.

17

Figure 4. Transient behavior of probe absorption for different values of (a)  2 , (b)  3 , and (c)  p , The selected parameters are, 2  3  3 3 , (a)  p  3  0 (b)  p   2  0 , (c) 3   2  0 .

Figure 5. Transient (a) and (b) steady- state behavior of probe absorption for different values of  2 . The selected parameters are 3   3 ,  p  2  3  0 , and the other parameters are the same as Fig. 2. Figure 6. Transient (a) and (b) steady- state behavior of probe absorption for different values of 3 . The selected parameters are 2   3 ,  p  2  3  0 , and the other parameters are the same as Fig. 2. Figure 7. Transient (a), (b) steady- state and population difference 11  22 (c) behavior of probe absorption for different values of

r . The selected parameters are 2  3   3 ,

 p  2  3  0 , and the other parameters are the same as Fig. 2.

Figure 8. Switching process for probe absorption for different values of  2 (a), 3 (b), and r (c). The selected parameters are 2  3   3 ,  p  2  3  0 , and the other parameters are the same as Fig. 2.

(a) (b)

Fig. 1 18

(a)

(b)

19

Fig. 2

(a)

(b)

20

Fig. 3

(a)

(b)

(c)

Fig. 4

21

(a)

(b)

Fig. 5

22

(a)

(b)

Fig. 6

23

(a)

(b)

(c)

Fig. 7

24

(a)

(b)

(c)

Fig. 8 Highlights

25



Transient behaviors of graphene structure is investigated.



The effect of various system parameters on transient of probe absorption is studied.



The switching times tom convert from a high- absorption case to the zero- absorption is discussed.

26