Effective terahertz signal detection via electromagnetically induced transparency in graphene

Transcription

1 Research Article Vol. 33, No. / February 06 / Journal of the Otical Society of America B 79 Effective terahertz signal detection via electromagnetically induced transarency in grahene SHAOPENG LIU, WEN-XING YANG,,, *ZHONGHU ZHU, AND RAY-KUANG LEE Deartment of Physics, Southeast University, Nanjing 0096, China Institute of Photonics Technologies, National Tsing-Hua University, Hsinchu 300, Taiwan *Corresonding author: wenxingyang@6.com Received 7 October 05; revised 8 December 05; acceted 9 December 05; osted December 05 (Doc. ID 5664); ublished February 06 We roose and analyze an efficient way to detect the terahertz signal in a magnetized grahene system via electromagnetically induced transarency. Such a scheme for THz signal detection mainly relies on the measurement of robe transmission sectra, in which the behaviors of a weak-robe transmission sectra can be controlled by switching on/off the THz signal radiation. Taking into account the tunable otical transition frequency between the Landau levels in grahene, our analytical results demonstrate that a broad frequency bandwidth of the THz signal radiation, ranging from 0.36 to.4 THz, can be insected and modulated by means of an external magnetic field. As a consequence, the roosed magnetized grahene system erforms a striking otential to utilize quantum interference in the design of otical solid-state devices. 06 Otical Society of America OCIS codes: (70.670) Coherent otical effects; (040.35) Far infrared or terahertz. htt://dx.doi.org/0.364/josab INTRODUCTION Grahene, consisting of a series of carbon atoms in the twodimensional hexagonal lattice, has attracted considerable scientific interest in its fascinating electronic and otical roerties during the ast few years. These electronic roerties originating from linear, massless disersion of electrons near the Dirac oint and the chiral character of electron states have motivated a number of recent theoretical investigations in grahene [ 3]. The magneto-otical roerties and thin grahite layers give rise to multile absortion eaks and articular selection rules between Landau levels (LLs) [4 7]. Subsequently, due to its unusual band structure and selection rules for the otical transitions, an intriguing otical nonlinearity in the infrared (IR) and terahertz region [5,6,8 3] has been exloited in the wide alications. For examle, Yao et al. have calculated the THz radiation ower generated by the four-wave mixing and stimulated Raman scattering rocesses in grahene [0]. In addition, several quantum theories about the frequency mixing effects and infrared solitons in grahene have been develoed [4 7]. From the viewoint of ractical alications, grahene may rovide a better access to the characterization and maniulation of otical roerties in the THz region. On the other hand, quantum interference in the form of electromagnetically induced transarency (EIT) has led to many significant research activities on otical communications and quantum information rocessing in both cold atom media and semiconductors [8 ]. This concet of quantum interference has also been extended to a variety of studies, such as controlling the grou velocity of light ulses [3,4], otical solitons [5,6], multiwave mixing rocess [7 9], otical bistability [30], and THz signal detection strategy [3 35]. In articular, the THz science is exected to have a wide range of alications in diverse fields, including biological imaging, long-distance detection of hazardous materials, and nondestructive testing [36 38]. Insired by the linear and nonlinear otical resonse of grahene in the IR and THz regions, we suggest a scheme for detecting and measuring the THz signal radiation by using EIT in grahene, which may oen u an avenue to exlore new availability for creating a comact and inexensive THz detector. In this aer, we study a novel otical method to detect the THz signal based on EIT in a quantized three-level grahene system under a strong magnetic field. Different from the most common THz technique using nonlinear multilication or uconversion of lower-frequency oscillators [39,40], our scheme based on quantum interference associating with one weak midinfrared robe light is analyzed for achieving the THz signal detection. Under suitable conditions, a strong absortion characteristic of the robe field aears in the absence of the THz signal, while the otical absortion of the robe field can be /6/ $5/0$ Otical Society of America

2 80 Vol. 33, No. / February 06 / Journal of the Otical Society of America B Research Article absolutely suressed with the THz signal switched on. The different resonse deends, resectively, on whether quantum destructive interference is quenched or well develoed so that we can examine the existence of the EIT transarency window and THz signal field. In addition, a time-deendent analysis erforms the robe ulse roagating with an ultraslow grou velocity (i.e., V g c) at a low-intensity domain. Motivated by establishing a ractical and effective aroach, the transmission sectra of the robe field is finally simulated by emloying the incident Gaussian-shaed robe ulse in our scheme of THz signal detection.. THEORETICAL MODEL AND BASIC EQUATIONS In this section, we roose a D grahene crystal structure with three energy levels in the resence of a strong magnetic field, as shown in Figs. (c). Because of the secial selection rules in resent grahene, the selected transitions are diole allowed between the aointed energy levels, i.e., Δjnj (n is the energy quantum number). Secially, the right-hand circularly (RHC) olarized hotons could be homogeneously alied to the condition of Δjnj, and the left-hand circularly (LHC) olarized hotons are simultaneously alied to the case Fig.. Landau levels (LLs) near the Dirac oint suerimosed on the linear electron disersion without the magnetic field E υ F jj. The magnetic field condenses the original states in the Dirac cone into discrete energies. Energy level diagram and otical transitions in grahene interacting with a weak robe ulse (with carrier frequency ω ) and a THz signal field (with carrier frequency ω TH ). The states ji, ji, and j3i corresond to the LLs with energy quantum numbers n ; ;, resectively. (c) Geometry of the system. The robe ulse and THz signal field are erendicularly incident on the single-layer grahene (the monolayer grahene is regarded as a erfect two-dimensional (D) crystal structure in the x y lane) laced in a magnetic field B, in which both two otical fields and magnetic field are along the z-axis. (d) Schematic of the two dressed states jai and jbi roduced by states ji and j3i couled with state ji in the resence of the THz signal. of Δjnj [5]. The alication of the external magnetic field results in the formation of discrete LLs, as shown in Fig.. It should be noted that the carrier frequencies of otical transition between adjacent LLs turn out to be in the IR or THz region ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for a magnetic field in the range of T: ħω c 36 B Tesla mev [6,0,4] (see Aendix A). We consider the electric field vector of the system that can be exressed as E e E ex iω t ik r c:c: and E TH e E TH ex iω THt ik TH r c:c:, where e e corresonds to the unit vector of the LHC (RHC) olarized basis, which can be noted as e ˆx iŷ ffiffi e ˆx iŷ ffiffi. In detail, the otical field of the LHC olarized comonent E E TH with the carrier frequency ω ω TH constructs the otical transition ji ji ji j3i, resectively. When the magnetic field is erendicularly alied in a single-layer grahene (in the x y lane), the otical transitions driven by the robe ulse and THz signal field generate the quantum interference and quantum coherence effects. In the interaction icture, with the rotating wave aroximation and the electric diole aroximation (see Aendix A), the total Hamiltonian of this system can be written as Ĥ I int Δ j3ih3j Δ Δ TH jihj Ω j3ihj Ω TH j3ihj h:c: ; () where the corresonding frequency detunings are given by Δ ε n ε n ħ ω and Δ TH ε n ε n ħ ω TH. The energy of the LLs for electrons ffiffiffiffiffi near the Dirac oint is denoted as ε n sgn n ħω c jnj n 0; ; with ω c ffiffiffi υf l c and l c ffiffiffiffiffiffiffiffiffiffiffiffi ħc eb reresenting the magnetic length. The corresonding Rabi frequencies for the relevant laser-driven intersubband transitions are reresented as Ω μ 3 e E ħ and Ω TH μ 3 e E TH ħ, in which μ mn hmj μjni e hmj rjni iħe ε n ε m hmjυ F σjni denotes the diole moments for the transition between states jmi jni. In the resent analysis, to obtain the absortion-disersion roerties of the grahene system, we first adot Liouville s equation ˆρ t i ħ Ĥ I int; ˆρ ˆR ˆρ. Here, ˆR ˆρ f ˆΓ; ˆρg f ˆΓ ˆρ ˆρ ˆΓg indicates incoherent relaxation, which may stem from disorder, interaction with honons and carrier carrier interactions. Moreover, the decay rate of the grahene is combined into the evolution equation by a relaxation matrix ˆΓ, which can be defined by hnj ˆΓjmi γ n δ nm. Subsequently, a standard time-evolution equation for the density matrix of Dirac electrons in grahene couled to the infrared laser fields can be calculated as follows, _ρ iω ρ 3 iω ρ 3 ; _ρ γ ρ iω TH ρ 3 iω TH ρ 3; _ρ 33 ρ 33 iω ρ 3 iω ρ 3 iω TH ρ 3 iω TH ρ 3 ; γ3 _ρ 3 iδ ρ 3 iω ρ ρ 33 iω TH ρ ; γ _ρ i Δ Δ TH ρ iω TH ρ 3 iω ρ 3 ; γ3 γ _ρ 3 iδ TH ρ 3 iω TH ρ ρ 33 iω ρ ; ()

3 Research Article Vol. 33, No. / February 06 / Journal of the Otical Society of America B 8 where γ i i ; 3 corresonds to the decay rate of states jii. And the steady-state solutions for Eq. () can be derived as ρ Ω Ω TH Ω TH d ; ρ 3 id Ω d 3 Ω TH d ; (3) d 3 with d γ i Δ Δ TH, d 3 iδ. Under an aroriate frame, we can substitute straightforwardly results of Eq. (3) into the slowly varying arts of the olarization of the weak robe ulse, i.e., P ε 0 χ ω E N μ 3 ρ 3 c:c:, with N being the electron concentration. The exression for the suscetibility χ ω can therefore be written as χ ω N μ 3 ε 0 ħ ρ3 id β Ω d d 3 Ω ; (4) TH where the arameter β is noted as β N μ 3 ε 0ħ. The real and imaginary arts of suscetibility χ ω are still linear with resect to disersion and absortion of the grahene system, resectively. Further, the absortion coefficient for the robe field couled to ji j3i is directly roortional to the imaginary art of Im ρ 3, while the of the robe field is sontaneously roortional to the imaginary art of suscetibility. For the roagation effect of the robe field, the electronic equations of motion must be simultaneously solved with Maxwell s equation in a self-consistent manner. Under the condition of slowly varying enveloe aroximation, Rabi frequency Ω of the robe field is governed by the following Maxwell s equation, Ω z Ω iα υ F t ρ 3 ; (5) where the arameter α N ω jμ 3 j 4ħε r υ F is the roagation constant of the robe field. When Ω in is assumed as the initial robe field at the entrance z 0, we arrive at the linearized results for roagation dynamics of the robe field at the outut z L: Ω out Ω in e α ρ3 L Im Ω : (6) Based on Eqs. (4) (6), the normalized of the robe field has the form T Ω out Ω in. It should be ointed out that the carrier frequencies of relevant LLs can be estimated by an amount ffiffi of the transition frequencies ω 3 ε n ffiffiffi ε n ħ ωc and ω 3 ε n ε n ħ ωc, and the carrier frequency ω c is determined by the magnetic field B. For the magnetic field u to 3 T, transition frequency ω c is of the order of ω c 0 4 s. In this scenario, ν THz (i.e., ω s ) is located within the mid-infrared region, while the ν THz (i.e., ω s ) belongs to the THz region. According to the numerical estimate based on [6,4], the decay rates can be estimated to be s and γ For the resent grahene system, the diole moment between the transition ji j3i has a magnitude of the order of j μ 3 j ħeυ F ε n ε n ffiffiffi B. The electron concentration can be estimated to be N 5 0 cm and the substrate dielectric constant turns out to be ε r 4.5 [4 44]. These arameter values rely on the samle quality and the substrate used in the exeriment [6,9,0,45,46]. 3. NUMERICAL RESULTS AND DISCUSSION It is well known that the frequencies involved in the otical transitions between adjacent LLs fall into the IR and THz region in the magnetized grahene and have a sensitive deendent on the ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi strength of the magnetic field via the relation ħω c 36 B Tesla. Within the above ractical arameter set, we lot the transmission sectra of the robe field as a function of the robe frequency and the magnetic field without and with including the THz signal field, as shown in Fig.. For the two cases, we investigate the transmission sectra of the robe field in the external magnetic field ranging from 0.0 to 0 T. When the THz signal radiation is off (i.e., jω TH j 0) in Fig., one can see that the only high absortion line aears in the center of the robe transmission sectra. And the center frequency of the robe absortion eak has a shift in the midinfrared region with the increase of the magnetic field. When the THz signal oens in Fig., the original high absortion line becomes an obvious transarency window between two high absortion lines. Thus, the THz signal radiation can be regarded as a switch to maniulate the robe transmission sectra with absortion or transarency. In order to simlify the above hysical icture, in Fig. 3 we show the transmission sectra of the robe fields versus the robe frequency, while keeing the external magnetic field fixed ν ν Fig.. Contour mas of transmission sectra of the robe field as a function of the robe frequency and the magnetic field. Without including THz signal, jω TH j 0; with including THz signal, jω TH j γ. Other arameters are Δ TH 0, γ s, γ, γ 0.05, jω j 0.05γ, L 0.4 nm, and α 00 μm. 5 Ω TH =0 Ω TH =γ ν ν Fig. 3. Transmission sectra of the robe field versus robe frequency for different THz signals with γ. Transmission sectra of the robe field versus robe frequency for different losses of grahene with jω TH j γ. Other arameters are B 3T, Δ TH 0, γ s, γ 0.05, jω j 0.05γ, L 0.4 nm, and α 00 μm. 5 =0.γ =0.5γ =5γ =0γ

4 8 Vol. 33, No. / February 06 / Journal of the Otical Society of America B Research Article (i.e., B 3T ). In the robe transmission sectra of the magnetized grahene system, a low transmission valley or high absortion eak aears around the resonant robe frequency (i.e., ν ω 3 π 38.4 THz) when the THz signal field switches off (i.e., jω TH j 0). Contrarily, the narrow EIT transarency window can be observed by switching on the THz signal radiation (i.e., jω TH j γ). In fact, these different otical resonses deend significantly on whether quantum destructive interference is quenched or well develoed [9,47,48] by modifying the strength of THz signal radiation. To investigate the influence of losses in grahene, we lot the robe transmission sectra versus robe frequency for different losses of grahene; see Fig. 3. The influence of losses in grahene is mainly embodied in the transarency extent of the EIT window, while keeing the otical laser fields fixed. That is, with the loss decreasing, the transarency window is more obvious, which strongly suorts our THz detection scheme. However, the large loss of grahene leads to an intensive disturbance for the EIT transarency window. As a result, a high-quality grahene with a weak loss may rovide a ractical hel to create the erfect transarency window. Furthermore, the above hysical henomenon can be also exlained by dressed states theory. Here, to achieve better EIT henomenon, we assume that the robe ulse and THz signal field satisfy the resonance condition (i.e., Δ 0 and Δ TH 0) and the relation jω TH j jω j. Simultaneously, the excited subband j3i is couled by the robe ulse to the ground state with μ 3, while it is intensively couled by the THz signal field to the subband ji with μ 3. There are two dress states jai and jbi, as shown in Fig. (d). The corresonding dress states under the one-hoton resonance condition can be exressed as jai ffiffiffi j3i ji ; jbi ffiffi j3i ji ; (7) with the energy eigenvalues of the two dressed states λ a;b Ω TH. When the frequency detuning of the robe field is tuned at Δ λ a and Δ λ b, two resonant excitations haen through the channels in the dressed state basis ji jai and ji jbi, which corresond to the two absortion eaks. At the osition of robe resonance Δ 0, the suerosition of two absortion aths creates the EIT transarency window. According to the above result, we have verified that the different otical resonse of the grahene occurs at the transarency window, so that we can detect the THz signal by measuring the strength of the robe transmission sectra. In the following discussion, we mainly investigate the otical roerties of the robe and THz signal fields in the magnetized grahene. First of all, we dislay transmission sectra of the robe field versus the THz signal intensity in Fig. 4. It can be seen that the transmission ratio increases monotonically with the increase of THz signal intensity until the aearance of the erfect EIT window. In comarison with the different losses of grahene, the smaller the loss of grahene is, the less THz signal intensity the erfect EIT system needs. For examle, to achieve the erfect EIT window, the THz signal intensity needs to reach at least 0 5 W cm hotons s er cm with the loss of grahene γ, while the THz signal intensity is 0 3 W cm hotons s er cm with the 5 =0.γ =γ =5γ THz signal intensity (W/cm ) Ω /γ TH Fig. 4. Transmission sectra of the robe field versus the THz signal intensity for different losses of grahene on a log log scale. Grou velocity of robe light as a function of amlitude jω TH j of THz signal field on a log log scale. Other arameters are B 3T, Δ 0, Δ TH 0, γ s, γ 0.05, jω j 0.05γ, L 0.4 nm, and α 00 μm. loss 0.γ. That is to say, we can get a further reduced THz signal intensity to achieve the erfect EIT transarency window and the THz detection scheme by decreasing the losses of the magnetized grahene system. Second, we lot the grou velocity as a function of THz signal amlitude jω TH j. The grou velocity of the robe field exhibits a ultraslow roagation regime and its order of magnitude reaches at c for the different losses of grahene. As a matter of fact, quantum destructive interference driven by the THz signal field modifies the disersive roerty of grahene, leading to the slow grou velocity. In view of a controllable normal disersion roerty in the narrow EIT window [4], a further reduced grou velocity can be obtained by choosing the related arameter values aroriately. For a direct insight into the effect of the THz signal frequency on the robe transmission sectra, we lot the contour mas of the robe transmission sectra as a function of the THz signal frequency and the magnetic field in Fig. 5. Here, we first assume that the robe field detuning is always zero. When the THz signal frequency aroaches to resonant transition frequency (i.e., ν TH ν 3 ω 3 π), the EIT window is exected to aear, so that the robe field safely asses through the grahene system. On the other hand, as the magnetic field is varying from 0.0 to 0 T, the carrier frequency ν 3 falls into the THz region ranging from 0.36 to.4 THz. According to that, Fig. 5 imlies that a broad bandwidth of the THz signal frequency can be accurately insected by controlling the external magnetic field. Interestingly enough, direct comarison of ν TH V g / c ν TH Fig. 5. Contour mas of transmission sectra of the robe field as a function of the THz signal frequency and the magnetic field. jω TH j γ 3; jω TH j γ. Other arameters are Δ 0, γ s, γ, γ 0.05, jω j 0.05γ, L 0.4 nm, and α 00 μm. =0.γ =γ =5γ 8 6 4

5 Research Article Vol. 33, No. / February 06 / Journal of the Otical Society of America B t/τ Ω TH /γ 3 0 the results of Figs. 5 and 5 illustrates that the transarency window can be obviously enlarged with the amlitude jω TH j increasing. U to now, we have demonstrated that the THz signal detection scheme can be achieved in the resent grahene under the external magnetic field based on quantum interference theory. Since the robe and THz signal fields are in the form of otics ulses, we also consider how to simulate a ractical aroach. Thus, we assume that a weak Gaussian-shaed robe light (i.e., Ω in e t τ ) with the ulse width τ sasses through the magnetized grahene structure. Now, the transmitted robe field can be rewritten as Ω out e t τ e α L Im ρ3 Ω. In Fig. 6, we show the results from numerical simulations with the different cases. For one case, in Fig. 6, it can be clearly seen that the transmissivity of the robe field significantly decreases with the amlitude of the THz signal field increasing until the aearance of a erfect EIT transarency window, which has been introduced in Fig. 4. For the other case, Fig. 6 describes that robe transmission sectra as a function of the dimensionless time t τ and the THz signal frequency. When the magnetic field remains fixed, the transarency window can be oened under the resonance condition of the signal field (i.e., the THz signal frequency equals the transition frequency between states ji and j3i). These observations are indeed similar to the revious research and rove the feasibility of this scheme. Secifically, similar to our Λ-tye model, the V- and cascade-tye schemes based on the three-state EIT can be suggested for detecting THz signal in a grahene system. 4. CONCLUSION In conclusion, we have erformed a scheme to realize THz signal detection under an external magnetic field due to the strong otical resonse of a grahene system in the IR and THz region. With the aid of the EIT and quantum destructive interference, the behaviors of the robe transmission sectra are determined by switching on/off THz signal radiation. Taking this theory and solving the couled Schrödinger Maxwell formula, we demonstrate that the magnetized grahene rovides the EIT transarency window in the resence of the THz signal field, in which the robe ulse roagates with an extremely slow grou velocity. Furthermore, the behavior of the robe ν TH Fig. 6. Surface lot of robe transmission sectra as a function of the dimensionless time t τ and amlitude jω TH j with Δ TH 0. Surface lot of robe transmission sectra as a function of the dimensionless time t τ and the THz signal frequency with jω TH j γ. Other arameters are B 3T, τ s, Δ 0, γ s, γ, γ 0.05γ, jω j 0.05γ, L 0.4 nm, and α 00 μm. t/τ 5 transmission sectra can be controlled by means of the otical absortion roerty of the grahene system so that a broad THz signal frequency bandwidth can be accurately insected. As a result, the above excellent erformances demonstrate that the grahene has a admirable otential to utilize its otical roerty in the design of otical detectors. APPENDIX A When the magnetic field is erendicularly alied in a single-layer grahene (in the z y lane), the effective-mass Hamiltonian [49 5] without external otical field can be written as 0 0 ˆπ x i ˆπ y 0 0 Ĥ 0 υ F B ˆπ x i ˆπ y ˆπ x i ˆπ y A ; 0 0 ˆπ x i ˆπ y 0 (A) where Fermi velocity υ F 3γ 0 ħa 0 6 m s is a band arameter with the nearest-neighbor hoing energy γ 0.8 ev and C-C sacing a.4a, ˆ π ˆ e A c reresents the generalized momentum oerator, ˆ is the electron momentum oerator, e is the electron charge, and A is the vector otential, which is equal to 0;Bx for a uniform magnetic field. In general, one can obtain the eigenenergies of discrete LLs for the magnetized grahene by solving the effective mass Schrödinger equations, i.e., Ĥ 0 Ψ εψ. In fact, the Hamiltonian near the K oint can be exressed as Ĥ 0 υ F ˆ σ ˆ π,whereˆ σ ˆσx ; ˆσ y is a vector of Pauli matrices. Then, the eigenfunction is secified by two quantum numbers n n 0; ; ; and the electron wave vector k y along the y direction [6,0,49], Ψ n;ky r C n ffiffiffi e yy sgn n i ik jnj φ jnj L i jnj ; (A) φ jnj with n 0 C n ffiffi n 0 ; (A3) and h φ jnj H jnj x l c k y l c ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x l i c k y lc jnj e ; (A4) jnj! π l c where l c ffiffiffiffiffiffiffiffiffiffiffiffi cħ eb is magnetic length and H n x is the Hermite olynomial. The eigenenergy can be calculated as ε n ffiffiffiffiffi sgn n ħω c jnj with ωc ffiffiffi υf l c. In comarison with LLs of a conventional D electron/hole system with a arabolic disersion, LLs in grahene are unequally ffiffiffi saced and their transition energies are roortional to B [6,9,0,45,46]. Combining the eigenenergy of the grahene system with the selected three energy levels in Fig., wecansimlifythesystem Hamiltonian without otical fields as Ĥ 0 ħε 3 j3ih3j ħε jihj ħε jihj: (A5) Considering the light matter interaction in the grahene system, the vector otential of the otical field A ot ic E ω

6 84 Vol. 33, No. / February 06 / Journal of the Otical Society of America B Research Article E E E TH is emloyed in the vector otential of the magnetic field in the generalized momentum oerator π in the Hamiltonian. The generated interaction Hamiltonian contained magnetized grahene and incident otical field can be obtained in the following form: Ĥ int υ F σ e A c ot : (A6) From the above equation, the interaction Hamiltonian does not include the momentum oerator, and it is only determined by the Pauli matrix vector and roortional to vector otential A ot. In addition, we insert the comlete set of states fj3i; ji; jig in the generated interaction Hamiltonian. Hence we obtain Ĥ int I υ F σ e A c ot I μ 3 E e iωt j3ihj μ 3 E TH e iωtht j3ihj h:c: ħω e iωt j3ihj ħω TH e iωtht j3ihj h:c:; (A7) where the comlete set of states is I Σjiihij (i ; ; 3) and i μ mn hmj μjni e hmj rjni iħe ε n ε m hmjυ F σjni denotes the diole moments for the transition between states jmi jni. The corresonding Rabi frequencies for the relevant laser-driven intersubband transitions are reresented as Ω μ 3 e E ħ and Ω TH μ 3 e E TH ħ. Then we can get the total Hamiltonian of the grahene system, i.e., Ĥ Ĥ 0 Ĥ int. To simlify this formula, we assume that the state ji is the zero otential reference. In the interaction icture, with the rotating wave aroximation and the electric diole aroximation, the total Hamiltonian can be exressed as (ħ ) Ĥ I int Δ j3ih3j Δ Δ TH jihj Ω j3ihj Ω TH j3ihj h:c: ; (A8) with the corresonding frequency detunings Δ ε n ε n ħ ω and Δ TH ε n ε n ħ ω TH. Funding. National Natural Science Foundation of China (NSFC) (374050, 6370); Fundamental Research Funds for the Central Universities (40R300); Scientific Research Foundation of Graduate School of Southeast University (YBJJ5). REFERENCES. R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. 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