Effective terahertz signal detection viaelectromagnetically induced transparency ingrapheneSHAOPENG LIU,1 WEN-XING YANG,1,2,* ZHONGHU ZHU,1 AND RAY-KUANG LEE2

1Department of Physics, Southeast University, Nanjing 210096, China2Institute of Photonics Technologies, National Tsing-Hua University, Hsinchu 300, Taiwan*Corresponding author: wenxingyang2@126.com

Received 27 October 2015; revised 8 December 2015; accepted 19 December 2015; posted 22 December 2015 (Doc. ID 252664);published 1 February 2016

We propose and analyze an efficient way to detect the terahertz (THz) signal in a magnetized graphene system viaelectromagnetically induced transparency. Such a scheme for THz signal detection mainly relies on the measure-ment of probe transmission spectra, in which the behaviors of a weak-probe transmission spectra can be con-trolled by switching on/off the THz signal radiation. Taking into account the tunable optical transition frequencybetween the Landau levels in graphene, our analytical results demonstrate that a broad frequency bandwidth ofthe THz signal radiation, ranging from 0.36 to 11.4 THz, can be inspected and modulated by means of an externalmagnetic field. As a consequence, the proposed magnetized graphene system performs a striking potential toutilize quantum interference in the design of optical solid-state devices. © 2016 Optical Society of America

OCIS codes: (270.1670) Coherent optical effects; (040.2235) Far infrared or terahertz.

http://dx.doi.org/10.1364/JOSAB.33.000279

1. INTRODUCTION

Graphene, consisting of a series of carbon atoms in the two-dimensional hexagonal lattice, has attracted considerable scien-tific interest in its fascinating electronic and optical propertiesduring the past few years. These electronic properties originat-ing from linear, massless dispersion of electrons near the Diracpoint and the chiral character of electron states have motivateda number of recent theoretical investigations in graphene [1–3].The magneto-optical properties and thin graphite layers giverise to multiple absorption peaks and particular selection rulesbetween Landau levels (LLs) [4–7]. Subsequently, due to itsunusual band structure and selection rules for the optical tran-sitions, an intriguing optical nonlinearity in the infrared (IR)and terahertz (THz) region [5,6,8–13] has been exploited inthe wide applications. For example, Yao et al. have calculatedthe THz radiation power generated by the four-wave mixingand stimulated Raman scattering processes in graphene [10].In addition, several quantum theories about the frequency mix-ing effects and infrared solitons in graphene have been devel-oped [14–17]. From the viewpoint of practical applications,graphene may provide a better access to the characterizationand manipulation of optical properties in the THz region.

On the other hand, quantum interference in the formof electromagnetically induced transparency (EIT) has led tomany significant research activities on optical communications

and quantum information processing in both cold atom mediaand semiconductors [18–22]. This concept of quantum inter-ference has also been extended to a variety of studies, such ascontrolling the group velocity of light pulses [23,24], opticalsolitons [25,26], multiwave mixing process [27–29], opticalbistability [30], and THz signal detection strategy [31–35]. Inparticular, the THz science is expected to have a wide range ofapplications in diverse fields, including biological imaging,long-distance detection of hazardous materials, and nonde-structive testing [36–38]. Inspired by the linear and nonlinearoptical response of graphene in the IR and THz regions, wesuggest a scheme for detecting and measuring the THz signalradiation by using EIT in graphene, which may open up anavenue to explore new availability for creating a compact andinexpensive THz detector.

In this paper, we study a novel optical method to detect theTHz signal based on EIT in a quantized three-level graphenesystem under a strong magnetic field. Different from the mostcommon THz technique using nonlinear multiplication or up-conversion of lower-frequency oscillators [39,40], our schemebased on quantum interference associating with one weak mid-infrared probe light is analyzed for achieving the THz signaldetection. Under suitable conditions, a strong absorption char-acteristic of the probe field appears in the absence of the THzsignal, while the optical absorption of the probe field can be

Research Article Vol. 33, No. 2 / February 2016 / Journal of the Optical Society of America B 279

0740-3224/16/020279-07$15/0$15.00 © 2016 Optical Society of America

absolutely suppressed with the THz signal switched on. Thedifferent response depends, respectively, on whether quantumdestructive interference is quenched or well developed so thatwe can examine the existence of the EIT transparency windowand THz signal field. In addition, a time-dependent analysisperforms the probe pulse propagating with an ultraslow groupvelocity (i.e., V g ≃ 10−3 ∼ 10−4c) at a low-intensity domain.Motivated by establishing a practical and effective approach,the transmission spectra of the probe field is finally simulatedby employing the incident Gaussian-shaped probe pulse in ourscheme of THz signal detection.

2. THEORETICAL MODEL AND BASICEQUATIONS

In this section, we propose a 2D graphene crystal structure withthree energy levels in the presence of a strong magnetic field, asshown in Figs. 1(a)–1(c). Because of the special selection rulesin present graphene, the selected transitions are dipole allowedbetween the appointed energy levels, i.e., Δjnj � �1 (n is theenergy quantum number). Specially, the right-hand circularly(RHC) polarized photons could be homogeneously appliedto the condition of Δjnj � −1, and the left-hand circularly(LHC) polarized photons are simultaneously applied to the case

of Δjnj � �1 [5]. The application of the external magneticfield results in the formation of discrete LLs, as shown inFig. 1(b). It should be noted that the carrier frequencies of op-tical transition between adjacent LLs turn out to be in the IR orTHz region for a magnetic field in the range of 0.01–10 T:ℏωc ≃ 36

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB�Tesla�

pmeV [6,10,41] (see Appendix A). We

consider the electric field vector of the system that can be ex-pressed as E p � e−E−

p exp�−iωpt � ikp · r� � c:c: and ETH �e−E−

TH exp�−iωTH t � ikTH · r� � c:c:, where e−�e�� corre-sponds to the unit vector of the LHC (RHC) polarized basis,which can be noted as e− � �x − iy�∕ ffiffiffi

2p �e� � �x � iy�∕ ffiffiffi

2p �.

In detail, the optical field of the LHC polarized component E−p

�E−TH � with the carrier frequency ωp �ωTH � constructs the op-

tical transition j1i↔j2i �j2i↔j3i�, respectively.When the magnetic field is perpendicularly applied in a

single-layer graphene (in the x − y plane), the optical transitionsdriven by the probe pulse and THz signal field generate thequantum interference and quantum coherence effects. In theinteraction picture, with the rotating wave approximation andthe electric dipole approximation (see Appendix A), the totalHamiltonian of this system can be written as

H Iint � Δpj3ih3j � �Δp − ΔTH �j2ih2j

− �Ωpj3ih1j � ΩTH j3ih2j � h:c:�; (1)

where the corresponding frequency detunings are given byΔp � �εn�2 − εn�−1�∕ℏ − ωp and ΔTH � �εn�2 − εn�1�∕ℏ − ωTH . The energy of the LLs for electrons near the Diracpoint is denoted as εn � sgn�n�ℏωc

ffiffiffiffiffiffijnj

p�n � 0;�1;�2�

with ωc �ffiffiffi2

pυF∕l c and l c �

ffiffiffiffiffiffiffiffiffiffiffiffiℏc∕eB

prepresenting the mag-

netic length. The corresponding Rabi frequencies for the rel-evant laser-driven intersubband transitions are represented asΩp � �μ31 · e−�E−

p∕ℏ and ΩTH � �μ32 · e−�E−TH∕ℏ, in which

μmn � hmjμjni � e · hmjrjni � iℏeεn−εm

hmjυF σjni denotes thedipole moments for the transition between states jmi↔jni.

In the present analysis, to obtain the absorption-dispersionproperties of the graphene system, we first adopt Liouville’sequation ∂ρ

∂t � − iℏ �H I

int; ρ� − R�ρ�. Here, R�ρ� � 12 fΓ; ρg �

12 fΓ ρ�ρ Γg indicates incoherent relaxation, which may stemfrom disorder, interaction with phonons and carrier–carrierinteractions. Moreover, the decay rate of the graphene is com-bined into the evolution equation by a relaxation matrix Γ,which can be defined by hnjΓjmi � γnδnm. Subsequently, astandard time-evolution equation for the density matrix ofDirac electrons in graphene coupled to the infrared laser fieldscan be calculated as follows,_ρ11 � iΩ

pρ31 − iΩpρ13;

_ρ22 � −γ2ρ22 � iΩTHρ32 − iΩTHρ23;

_ρ33 � −γ3ρ33 � iΩpρ13 − iΩpρ31 � iΩ

THρ23 − iΩTHρ32;

_ρ31 � −�γ32� iΔp

�ρ31 � iΩp�ρ11 − ρ33� � iΩTHρ21;

_ρ21 � −�γ22� i�Δp − ΔTH �

�ρ21 � iΩ

THρ31 − iΩpρ23;

_ρ32 � −�γ3 � γ2

2� iΔTH

�ρ32 � iΩTH �ρ22 − ρ33� � iΩpρ12;

(2)

Fig. 1. (a) Landau levels (LLs) near the Dirac point superimposedon the linear electron dispersion without the magnetic field E ��υF jpj. The magnetic field condenses the original states in the Diraccone into discrete energies. (b) Energy level diagram and optical tran-sitions in graphene interacting with a weak probe pulse (with carrierfrequency ωp) and a THz signal field (with carrier frequency ωTH ).The states j1i, j2i, and j3i correspond to the LLs with energy quan-tum numbers n � −1; 1; 2, respectively. (c) Geometry of the system.The probe pulse and THz signal field are perpendicularly incident onthe single-layer graphene (the monolayer graphene is regarded as a per-fect two-dimensional (2D) crystal structure in the x − y plane) placedin a magnetic field B, in which both two optical fields and magneticfield are along the z-axis. (d) Schematic of the two dressed states jaiand jbi produced by states j2i and j3i coupled with state j1i in thepresence of the THz signal.

280 Vol. 33, No. 2 / February 2016 / Journal of the Optical Society of America B Research Article

where γi�i � 2; 3� corresponds to the decay rate of states jii.And the steady-state solutions for Eq. (2) can be derived as

ρ21 �−ΩpΩTH

Ω2TH � d 21d 31

; ρ31 �−id 21Ωp

Ω2TH � d 21d 31

; (3)

with d 21 � −�γ22 � i�Δp − ΔTH ��, d 31 � −�γ32 � iΔp�.Under an appropriate frame, we can substitute straightfor-

wardly results of Eq. (3) into the slowly varying parts of thepolarization of the weak probe pulse, i.e., P � ε0χ�ωp�Ep �N �μ31ρ31 � c:c:�, with N being the electron concentration.The expression for the susceptibility χ�ωp� can therefore bewritten as

χ�ωp� �Nμ231ε0ℏ

·ρ31Ωp

� β−id 21

d 21d 31 �Ω2TH

; (4)

where the parameter β is noted as β � Nμ231∕ε0ℏ. The real andimaginary parts of susceptibility χ�ωp� are still linear with re-spect to dispersion and absorption of the graphene system, re-spectively. Further, the absorption coefficient for the probe fieldcoupled to j1i↔j3i is directly proportional to the imaginarypart of Im�ρ31�, while the transmission coefficient of the probefield is spontaneously proportional to the imaginary part of sus-ceptibility. For the propagation effect of the probe field, theelectronic equations of motion must be simultaneously solvedwith Maxwell’s equation in a self-consistent manner. Under thecondition of slowly varying envelope approximation, Rabi fre-quency Ωp of the probe field is governed by the followingMaxwell’s equation,

∂Ωp

∂z� 1

υF

∂Ωp

∂t� iαpγ3ρ31; (5)

where the parameter αp � Nωpjμ31j24ℏεrυF γ3

is the propagation constantof the probe field. When Ωin

p is assumed as the initial probefield at the entrance z � 0, we arrive at the linearized resultsfor propagation dynamics of the probe field at the outputz � L:

Ωoutp � Ωin

p e−αpLγ3 Im

�ρ31Ωp

�: (6)

Based on Eqs. (4)–(6), the normalized transmission coeffi-cient of the probe field has the form T ≡Ωout

p ∕Ωinp .

It should be pointed out that the carrier frequencies ofrelevant LLs can be estimated by an amount of the transi-tion frequencies ω31 � �εn�2 − εn�−1�∕ℏ � � ffiffiffi

2p � 1�ωc and

ω32 � �εn�2 − εn�1�∕ℏ � � ffiffiffi2

p− 1�ωc , and the carrier fre-

quency ωc is determined by the magnetic field B. For themagnetic field up to 3 T, transition frequency ωc is of theorder of ωc ≃ 1014 s−1. In this scenario, ν31 ≃ 38.4 THz (i.e.,ω31 ≃ 2.41 × 1014 s−1) is located within the mid-infrared re-gion, while the ν32 ≃ 6.59 THz (i.e., ω32 ≃ 4.14 × 1013 s−1)belongs to the THz region. According to the numerical esti-mate based on [6,41], the decay rates can be estimated to beγ3 � 3 × 1013 s−1 and γ2 � 0.05γ3. For the present graphenesystem, the dipole moment between the transition j1i↔j3i hasa magnitude of the order of jμ31j ∼ ℏeυF∕�εn�2 − εn�−1� ∝1∕

ffiffiffiB

p. The electron concentration can be estimated to be

N ≃ 5 × 1012 cm−2 and the substrate dielectric constant turnsout to be εr ≃ 4.5 [42–44]. These parameter values rely

on the sample quality and the substrate used in the experi-ment [6,9,10,45,46].

3. NUMERICAL RESULTS AND DISCUSSION

It is well known that the frequencies involved in the opticaltransitions between adjacent LLs fall into the IR and THz re-gion in the magnetized graphene and have a sensitive depen-dent on the strength of the magnetic field via the relationℏωc ≃ 36

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB�Tesla�

p. Within the above practical parameter

set, we plot the transmission spectra of the probe field as a func-tion of the probe frequency and the magnetic field without andwith including the THz signal field, as shown in Fig. 2. For thetwo cases, we investigate the transmission spectra of the probefield in the external magnetic field ranging from 0.01 to 10 T.When the THz signal radiation is off (i.e., jΩTH j � 0) inFig. 2(a), one can see that the only high absorption line appearsin the center of the probe transmission spectra. And the centerfrequency of the probe absorption peak has a shift in the mid-infrared region with the increase of the magnetic field. Whenthe THz signal opens in Fig. 2(b), the original high absorptionline becomes an obvious transparency window between twohigh absorption lines. Thus, the THz signal radiation can beregarded as a switch to manipulate the probe transmission spec-tra with absorption or transparency.

In order to simplify the above physical picture, in Fig. 3(a)we show the transmission spectra of the probe fields versus theprobe frequency, while keeping the external magnetic field fixed

magnetic field B (T)

ν p (T

Hz)

(a)

0.01 2 4 6 8 101

20

40

60

80

0.86

0.88

0.9

0.92

0.94

0.96

0.98

magnetic field B (T)

ν p (T

Hz)

(b)

0.01 2 4 6 8 101

20

40

60

80

0.86

0.88

0.9

0.92

0.94

0.96

0.98

Fig. 2. Contour maps of transmission spectra of the probe fieldas a function of the probe frequency and the magnetic field.(a) Without including THz signal, jΩTH j � 0; (b) with includingTHz signal, jΩTH j � γ. Other parameters are ΔTH � 0, γ �3 × 1013 s−1, γ3 � γ, γ2 � 0.05γ3, jΩpj � 0.05γ, L � 0.4 nm, andαp � 200 μm−1.

20 30 40 500.85

0.9

0.95

1

νp (THz)

tran

smis

sion

coe

ffici

ent

(a)

|ΩTH

|=0

|ΩTH

|=γ

20 30 40 500.85

0.9

0.95

1

νp (THz)

tran

smis

sion

coe

ffici

ent

(b)

γ3=0.1γ

γ3=0.5γ

γ3=5γ

γ3=10γ

Fig. 3. (a) Transmission spectra of the probe field versus probefrequency for different THz signals with γ3 � γ. (b) Transmissionspectra of the probe field versus probe frequency for different lossesof graphene with jΩTH j � γ. Other parameters are B � 3T , ΔTH �0, γ � 3 × 1013 s−1, γ2 � 0.05γ3, jΩpj � 0.05γ, L � 0.4 nm, andαp � 200 μm−1.

Research Article Vol. 33, No. 2 / February 2016 / Journal of the Optical Society of America B 281

(i.e., B � 3T ). In the probe transmission spectra of the mag-netized graphene system, a low transmission valley or highabsorption peak appears around the resonant probe frequency(i.e., νp � ω31∕2π � 38.42 THz) when the THz signal fieldswitches off (i.e., jΩTH j � 0). Contrarily, the narrow EITtransparency window can be observed by switching on theTHz signal radiation (i.e., jΩTH j � γ). In fact, these differentoptical responses depend significantly on whether quantum de-structive interference is quenched or well developed [19,47,48]by modifying the strength of THz signal radiation. To inves-tigate the influence of losses in graphene, we plot the probetransmission spectra versus probe frequency for different lossesof graphene; see Fig. 3(b). The influence of losses in graphene ismainly embodied in the transparency extent of the EIT win-dow, while keeping the optical laser fields fixed. That is, withthe loss decreasing, the transparency window is more obvious,which strongly supports our THz detection scheme. However,the large loss of graphene leads to an intensive disturbance forthe EIT transparency window. As a result, a high-quality gra-phene with a weak loss may provide a practical help to createthe perfect transparency window. Furthermore, the abovephysical phenomenon can be also explained by dressed statestheory. Here, to achieve better EIT phenomenon, we assumethat the probe pulse and THz signal field satisfy the resonancecondition (i.e., Δp � 0 and ΔTH � 0) and the relationjΩTH j ≫ jΩpj. Simultaneously, the excited subband j3i iscoupled by the probe pulse to the ground state with μ31, whileit is intensively coupled by the THz signal field to the subbandj2i with μ32. There are two dress states jai and jbi, as shown inFig. 1(d). The corresponding dress states under the one-photonresonance condition can be expressed as

jai � 1ffiffiffi2

p �j3i � j2i�; jbi � 1ffiffiffi2

p �j3i − j2i�; (7)

with the energy eigenvalues of the two dressed statesλa;b � �ΩTH . When the frequency detuning of the probefield is tuned at Δp � λa and Δp � λb, two resonant excita-tions happen through the channels in the dressed state basisj1i → jai and j1i → jbi, which correspond to the two absorp-tion peaks. At the position of probe resonance Δp � 0, thesuperposition of two absorption paths creates the EIT transpar-ency window.

According to the above result, we have verified that thedifferent optical response of the graphene occurs at the trans-parency window, so that we can detect the THz signal by meas-uring the strength of the probe transmission spectra. In thefollowing discussion, we mainly investigate the optical proper-ties of the probe and THz signal fields in the magnetized gra-phene. First of all, we display transmission spectra of the probefield versus the THz signal intensity in Fig. 4(a). It can be seenthat the transmission ratio increases monotonically with the in-crease of THz signal intensity until the appearance of the per-fect EIT window. In comparison with the different losses ofgraphene, the smaller the loss of graphene is, the less THz signalintensity the perfect EIT system needs. For example, to achievethe perfect EIT window, the THz signal intensity needs toreach at least 105 W∕cm2 ∼ 6.25 × 1023 photons∕s per cm2

with the loss of graphene γ3 � γ, while the THz signal inten-sity is 103 W∕cm2 ∼ 6.25 × 1021 photons∕s per cm2 with the

loss γ3 � 0.1γ. That is to say, we can get a further reduced THzsignal intensity to achieve the perfect EIT transparency windowand the THz detection scheme by decreasing the losses of themagnetized graphene system. Second, we plot the group veloc-ity as a function of THz signal amplitude jΩTH j. The groupvelocity of the probe field exhibits a ultraslow propagation re-gime and its order of magnitude reaches at 10−3–10−4 c for thedifferent losses of graphene. As a matter of fact, quantum de-structive interference driven by the THz signal field modifiesthe dispersive property of graphene, leading to the slow groupvelocity. In view of a controllable normal dispersion property inthe narrow EIT window [24], a further reduced group velocitycan be obtained by choosing the related parameter valuesappropriately.

For a direct insight into the effect of the THz signal fre-quency on the probe transmission spectra, we plot the contourmaps of the probe transmission spectra as a function of the THzsignal frequency and the magnetic field in Fig. 5. Here, we firstassume that the probe field detuning is always zero. When theTHz signal frequency approaches to resonant transition fre-quency (i.e., νTH � ν32 � ω32∕2π), the EIT window is ex-pected to appear, so that the probe field safely passes throughthe graphene system. On the other hand, as the magnetic fieldis varying from 0.01 to 10 T, the carrier frequency ν32 falls intothe THz region ranging from 0.36 to 11.4 THz. According tothat, Fig. 5 implies that a broad bandwidth of the THz signalfrequency can be accurately inspected by controlling the exter-nal magnetic field. Interestingly enough, direct comparison of

101

102

103

104

105

106

0.85

0.9

0.95

1

THz signal intensity (W/cm2)

tran

smis

sion

coe

ffici

ent (a)

γ3=0.1γ

γ3=γ

γ3=5γ

0.1 1 1010

−5

10−4

10−3

10−2

10−1

|ΩTH

|/γ

Vg /

c

(b)

γ3=0.1γ

γ3=γ

γ3=5γ

Fig. 4. (a) Transmission spectra of the probe field versus the THzsignal intensity for different losses of graphene on a log–log scale.(b) Group velocity of probe light as a function of amplitude jΩTH jof THz signal field on a log–log scale. Other parameters areB � 3T , Δp � 0, ΔTH � 0, γ � 3 × 1013 s−1, γ2 � 0.05γ3, jΩpj �0.05γ, L � 0.4 nm, and αp � 200 μm−1.

magnetic field B (T)

ν TH

(T

Hz)

(a)

0.01 2 4 6 8 10

1

5

10

15

0.86

0.88

0.9

0.92

0.94

0.96

0.98

magnetic field B (T)

ν TH

(T

Hz)

(b)

0.01 2 4 6 8 10

1

5

10

15

0.9

0.92

0.94

0.96

0.98

Fig. 5. Contour maps of transmission spectra of the probe field asa function of the THz signal frequency and the magnetic field.(a) jΩTH j � γ∕3; (b) jΩTH j � γ. Other parameters are Δp � 0,γ � 3 × 1013 s−1, γ3 � γ, γ2 � 0.05γ3, jΩpj � 0.05γ, L � 0.4 nm,and αp � 200 μm−1.

282 Vol. 33, No. 2 / February 2016 / Journal of the Optical Society of America B Research Article

the results of Figs. 5(a) and 5(b) illustrates that the transparencywindow can be obviously enlarged with the amplitude jΩTH jincreasing.

Up to now, we have demonstrated that the THz signal de-tection scheme can be achieved in the present graphene underthe external magnetic field based on quantum interferencetheory. Since the probe and THz signal fields are in the formof optics pulses, we also consider how to simulate a practicalapproach. Thus, we assume that a weak Gaussian-shaped probelight (i.e., Ωin

p � e−t2∕τ2 ) with the pulse width τ � 1 ps passesthrough the magnetized graphene structure. Now, the transmit-

ted probe field can be rewritten as Ωoutp � e−t2∕τ2e−αpLγ3 Im�

ρ31Ωp �.

In Fig. 6, we show the results from numerical simulations withthe different cases. For one case, in Fig. 6(a), it can be clearlyseen that the transmissivity of the probe field significantly de-creases with the amplitude of the THz signal field increasinguntil the appearance of a perfect EIT transparency window,which has been introduced in Fig. 4(a). For the other case,Fig. 6(b) describes that probe transmission spectra as a functionof the dimensionless time t∕τ and the THz signal frequency.When the magnetic field remains fixed, the transparency win-dow can be opened under the resonance condition of the signalfield (i.e., the THz signal frequency equals the transition fre-quency between states j2i and j3i). These observations are in-deed similar to the previous research and prove the feasibility ofthis scheme. Specifically, similar to our Λ-type model, theV- and cascade-type schemes based on the three-state EIT canbe suggested for detecting THz signal in a graphene system.

4. CONCLUSION

In conclusion, we have performed a scheme to realize THz sig-nal detection under an external magnetic field due to the strongoptical response of a graphene system in the IR and THz re-gion. With the aid of the EIT and quantum destructive inter-ference, the behaviors of the probe transmission spectra aredetermined by switching on/off THz signal radiation. Takingthis theory and solving the coupled Schrödinger–Maxwell for-mula, we demonstrate that the magnetized graphene providesthe EIT transparency window in the presence of the THz signalfield, in which the probe pulse propagates with an extremelyslow group velocity. Furthermore, the behavior of the probe

transmission spectra can be controlled by means of the opticalabsorption property of the graphene system so that a broadTHz signal frequency bandwidth can be accurately inspected.As a result, the above excellent performances demonstrate thatthe graphene has a admirable potential to utilize its opticalproperty in the design of optical detectors.

APPENDIX A

When the magnetic field is perpendicularly applied in asingle-layer graphene (in the z − y plane), the effective-massHamiltonian [49–51] without external optical field can bewritten as

H 0 � υF

0BB@

0 πx − iπy 0 0πx � iπy 0 0 0

0 0 0 πx � iπy0 0 πx − iπy 0

1CCA;

(A1)

where Fermi velocity υF � 3γ0∕2ℏa ≈ 106 m∕s is a bandparameter with the nearest-neighbor hopping energy γ0 ∼2.8 eV and C-C spacing a � 1.42A°, ˆπ � ˆp� eA∕c representsthe generalized momentum operator, ˆp is the electron momen-tum operator, e is the electron charge, and A is the vector po-tential, which is equal to �0; Bx� for a uniform magnetic field.In general, one can obtain the eigenenergies of discrete LLs forthe magnetized graphene by solving the effective mass Schrödingerequations, i.e., H 0Ψ � εΨ. In fact, the Hamiltonian near the Kpoint can be expressed as H 0 � υF ˆσ · ˆπ, where ˆσ � �σx ; σy� is avector of Pauli matrices. Then, the eigenfunction is specified bytwo quantum numbers n�n � 0;�1;�2;…� and the electronwave vector ky along the y direction [6,10,49],

Ψn;ky �r� �CnffiffiffiL

p e�−ikyy��sgn�n�ijnj−1φjnj−1

ijnjφjnj

�; (A2)

with

Cn �1�n � 0�1ffiffi2

p �n ≠ 0� ; (A3)

and

φjnj �H jnj��x − l 2c ky�∕l c�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2jnjjnj! ffiffiffiπ

pl c

p e

h−12

�x−l2c ky

l c

�2i; (A4)

where l c �ffiffiffiffiffiffiffiffiffiffiffifficℏ∕eB

pis magnetic length and Hn�x� is the

Hermite polynomial. The eigenenergy can be calculated as εn �sgn�n�ℏωc

ffiffiffiffiffiffijnj

pwith ωc �

ffiffiffi2

pυF∕l c . In comparison with LLs

of a conventional 2D electron/hole system with a parabolicdispersion, LLs in graphene are unequally spaced and theirtransition energies are proportional to

ffiffiffiB

p[6,9,10,45,46].

Combining the eigenenergy of the graphene system with the se-lected three energy levels in Fig. 1(b), we can simplify the systemHamiltonian without optical fields as

H 0 � ℏε3j3ih3j � ℏε2j2ih2j � ℏε1j1ih1j: (A5)

Considering the light–matter interaction in the graphene sys-tem, the vector potential of the optical field Aopt � icE∕ω

00.5

1

−30

30

0.5

1

|ΩTH|/γt/τ

(a)

tran

smis

sion

coe

ffici

ent

0 5 10 15

−30

30

0.5

1

νTH (THz)t/τ

(b)

tran

smis

sion

coe

ffici

ent

Fig. 6. (a) Surface plot of probe transmission spectra as a functionof the dimensionless time t∕τ and amplitude jΩTH j with ΔTH � 0.(b) Surface plot of probe transmission spectra as a function of thedimensionless time t∕τ and the THz signal frequency withjΩTH j � γ. Other parameters are B � 3T , τ � 1 ps, Δp � 0,γ � 3 × 1013 s−1, γ3 � γ, γ2 � 0.05γ, jΩpj � 0.05γ, L � 0.4 nm,and αp � 200 μm−1.

Research Article Vol. 33, No. 2 / February 2016 / Journal of the Optical Society of America B 283

�E � E p � ETH � is employed in the vector potential of themagnetic field in the generalized momentum operator π inthe Hamiltonian. The generated interaction Hamiltonian con-tained magnetized graphene and incident optical field canbe obtained in the following form:

H int � υF σ ·ecAopt: (A6)

From the above equation, the interaction Hamiltonian doesnot include the momentum operator, and it is only determinedby the Pauli matrix vector and proportional to vector potentialAopt. In addition, we insert the complete set of states fj3i;j2i; j1ig in the generated interaction Hamiltonian. Hencewe obtain

H int � I · υF σ ·ecAopt · I

� −μ31E−pe−iωpt j3ih1j − μ32E−

TH e−iωTH t j3ih2j � h:c:

� −ℏΩpe−iωpt j3ih1j − ℏΩTH e−iωTH t j3ih2j � h:c:; (A7)

where the complete set of states is I � Σijiihij (i � 1; 2; 3) and

μmn � hmjμjni � e · hmjrjni � iℏeεn−εm

hmjυF σjni denotes thedipole moments for the transition between states jmi↔jni.The corresponding Rabi frequencies for the relevant laser-drivenintersubband transitions are represented as Ωp � �μ31 · e−�E−p∕ℏ and ΩTH � �μ32 · e−�E−

TH∕ℏ.Then we can get the total Hamiltonian of the graphene

system, i.e., H � H 0 � H int. To simplify this formula, we as-sume that the state j1i is the zero potential reference. In theinteraction picture, with the rotating wave approximationand the electric dipole approximation, the total Hamiltoniancan be expressed as (ℏ � 1)

H Iint � Δpj3ih3j � �Δp − ΔTH �j2ih2j

− �Ωpj3ih1j �ΩTH j3ih2j � h:c:�; (A8)

with the corresponding frequency detunings Δp � �εn�2−εn�−1�∕ℏ − ωp and ΔTH � �εn�2 − εn�1�∕ℏ − ωTH .

Funding. National Natural Science Foundation of China(NSFC) (11374050, 61372102); Fundamental ResearchFunds for the Central Universities (2242012R30011);Scientific Research Foundation of Graduate School ofSoutheast University (YBJJ1522).

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