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Optical bistability induced by spin–orbit coupling in the carbon-nanotube quantum dots WEI LIU, HONGJUN ZHANG,* HUI SUN, QIAOLIN ZHANG,
AND
DANDAN WANG
School of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, China *Corresponding author: [email protected] Received 30 October 2015; revised 3 January 2016; accepted 3 January 2016; posted 5 January 2016 (Doc. ID 253045); published 10 February 2016
We theoretically investigate steady-state behaviors of carbon nanotube quantum dots with spin–orbit coupling in a unidirectional ring cavity. Our results show that the spin–orbit coupling can induce optical bistability. In addition to the existence of spin–orbit coupling, switching optical bistability to optical multistability or vice versa is realized by adjusting detuning of the probe field. Our results show that the carbon nanotube quantum dots will possibly be a promising candidate for realizing all-optical switching. © 2016 Optical Society of America OCIS codes: (190.1450) Bistability; (190.3270) Kerr effect; (190.5970) Semiconductor nonlinear optics including MQW. http://dx.doi.org/10.1364/AO.55.001090
1. INTRODUCTION During the past several decades, optical bistability (OB) has attracted interest of many researchers due to its broad applications in all-optical switching [1], logic functions [2], all-optical transistor [3], and so on. OB usually occurs with a nonlinear medium placed in a unidirectional ring cavity [4–7], which constructs important feedback. It has been studied widely in atomic systems [8–14] and semiconducting schemes [15–18]. Harshawardhan and Agarwal [8] have reported that electromagnetically induced transparency (EIT) can lower the threshold of OB. The squeezed vacuum interacting with atoms [9–12] has been shown to have a remarkable effect on OB. The phase control has been used to manipulate the threshold of OB. For instance, in a closed system the threshold of OB has been controlled by the relative phase [13,17]. Particularly, Cheng et al. have found that the relative phase can also manipulate the threshold of OB with the existence of spontaneously generated coherence, although the system is not closed [14]. In addition, recent publications have shown that OB depends on Fano interference [15,16] and tunneling-induced transparency [18] in semiconducting schemes. Furthermore, optical multistability (OM) has been theoretically predicted [19] and experimentally carried out [7]. Recently, a single graphene layer has been given much attention for its zero-gap property in Dirac points [20], linear dispersion in the vicinity of Dirac points [21], and strong optical nonlinearity [22,23]. Peres et al. [24] have found that the inherent optical nonlinearity of graphene can lead to OB. Besides, some publications have predicted that graphene derivatives are potential candidates for spin-based applications [25–27] because of their long intrinsic coherence times and 1559-128X/16/051090-05$15/0$15.00 © 2016 Optical Society of America
the absence of nuclear spin [28], i.e., carbon nanotube (CNT). Some theories indicate that there is a strong spin–orbit coupling in CNT [29–31], although the spin–orbit coupling is weak in a single graphene [30]. Subsequently, it has been experimentally demonstrated that an electron’s orbit and its spin are strongly coupled [32]. The coupling supports parallel alignment of the orbital and spin magnetic moments for electrons and antiparallel alignment for holes. On the basis of the spin–orbit coupling, Galland and Imamoglu [33] have demonstrated theoretically all-optical manipulation of electron spins with a high fidelity. Moreover, the spin–orbit coupling, which leads to energy level shift, can effectively overcome group-velocity mismatching so that the giant cross-Kerr nonlinearity and coherent entanglement can be realized [34]. In this paper, motivated by spin–orbit coupling of CNT, we investigate the bistable behaviors of CNT quantum dots with spin–orbit coupling in a unidirectional ring cavity. We find that OB is induced by the spin–orbit coupling and can be manipulated by intensity and detuning of driving field. Beside, our results show OB is able to be switched to OM or vice versa with the spin–orbit coupling by adjusting the detuning of probe field. This paper is arranged as follows. The following section gives model and master equations, which describe the dynamic evolution of electrons with spin–orbit coupling; numerical results and explanations are presented in Section 3. Section 4 is the conclusion. 2. MODEL AND MASTER EQUATIONS As shown in Fig. 1(a), a quantum dot (QD) consisting of semiconducting CNT with a diameter d ∼ 1.2 nm is considered. The QD has a lowest optical transition in the near-infrared
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Fig. 1. (a) Nanotube with two top gates. The probe and driving fields are applied perpendicular to the CNT axis. (b) Energy diagram of the lowest electron and hole states in a nanotube quantum dot versus the applied axial magnetic field B ∥ .
(∼1500 nm). As shown in Fig. 1(b), the zero-field fourfold degeneracy [35,36] is split into two pairs of doublet states due to the existence of strong spin–orbit coupling. Two pairs of doublets, holding energy differences [33] Δeso ≈ 1.5 meV and Δhso ≈ 0.9 meV, have parallel and antiparallel spin and orbital magnetic moments. Here, SO represents spin-orbit coupling and eh is electron (hole). The doublet states can be separated by applying a magnetic field B ∥ parallel to the CNT axis because of Zeeman effects. To distinguish different states, we label the states with combinations of U D, eh, and arrows in Fig. 1(b) where U D signifies the positive (negative) orbital magnetic moment, eh represents electron (hole), and arrows stand for the projection of the spin along the CNT axis. Taken into account momentum conservation, the types of U → U and D → D are allowable optical transitions. In addition, the up and down electron (hole) spin states are coherently coupled by the magnetic field orthogonal to the CNT axis [34,35]. For convenience of calculating, Fig. 1(b) is converted to Fig. 2(a) equivalently. The probe field with angular frequency ν1 and driving field with angular frequency ν2 are applied to couple with j1i↔j3i and j2i↔j4i, respectively. At the same time, level j3i and level j4i are coupled by a transverse
magnetic field with angular frequency ν3, which is orthogonal to the CNT axis. Here, Δ1 ω31 − ν1 , Δ2 ω42 − ν2 , and Δ3 ω34 − ν3 are the detunings of probe field, driving field, and transverse magnetic field, respectively, in which ωij is the energy difference between jii and jji. For analyzing solely the effect of spin–orbit coupling, we introduce two parameters, Δ1 ω42 − ν1 and ΔSO ΔhSO ΔeSO , which is the energy shift caused by spin–orbit coupling. Then, Fig. 2(a) can be transformed into Fig. 2(b). Under electric-dipole and rotating wave approximations, the interaction Hamiltonian of Fig. 2(b) can be given as follows (ℏ 1): H int Δ1 Δso − Δ2 − Δ3 j2ih2j Δ1 Δso j3ih3j Δ1 Δso − Δ3 j4ih4j − Ω1 j1ih3j Ω2 j2ih4j Ωj4ih3j h:c:;
(1)
in which h.c. is Hermitian conjugation. Ω1 E 12ℏμ31 and Ω2 E 2 μ42 2ℏ are half of Rabi frequencies of probe field and driving field, where μnm denotes the dipole moment for the transition between jni and jmi. Magnetic Rabi frequency Ω is denoted by gμB B ⊥ 2ℏ , whereby Lande factor g ≈ 2 is for the electron and hole,
Fig. 2. (a) Simple diagram is applied to illustrate that CNT interacts with optical fields and magnetic field. (b) Adapted diagram for highlighting spin-orbit coupling.
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and μB is the Bohr magneton. Because of the coupling resonantly the transition between state j3i and state j4i, the detuning of the transverse magnetic field (Δ3 ) is zero. Consequently, the matrix equations are given as follows: ∂ ρ iΩ1 ρ31 − iΩ1 ρ13 γ 41 ρ44 γ 31 ρ33 ; (2-1) ∂t 11 ∂ ρ iΩ2 ρ42 − iΩ2 ρ24 γ 32 ρ33 γ 42 ρ44 ; (2-2) ∂t 22 ∂ ρ iΩ1 ρ13 − iΩ1 ρ31 iΩρ43 − iΩ ρ34 − γ 31 ρ33 ∂t 33 − γ 32 ρ33 ; (2-3) ∂ ρ −iΩ2 ρ42 iΩ2 ρ24 iΩ ρ34 − iΩρ43 ∂t 44 − γ 41 ρ44 − γ 42 ρ44 ; (2-4) ∂ iγ ρ21 −iΩ1 ρ23 iΩ2 ρ41 − i Δ1 Δso − Δ2 − 41 ρ21 ; 2 ∂t (2-5)
∂ ρ iΩ1 ρ11 − ρ33 iΩρ41 ∂t 31 iγ γ 32 ρ31 ; (2-6) − i Δ1 Δso − 31 2 ∂ ρ iΩ1 ρ12 iΩρ42 − iΩ2 ρ34 ∂t 32 iγ γ 32 γ 21 ρ32 ; (2-7) − i Δ1 Δso − 31 2 ∂ ρ −iΩ1 ρ43 iΩ2 ρ21 iΩρ31 ∂t 41 iγ γ 41 ρ41 ; − i Δ1 Δso − 42 (2-8) 2 ∂ ρ iΩ2 ρ22 − ρ44 iΩ ρ32 ∂t 42 iγ γ 41 γ 21 ρ42 ; − i Δ2 − 42 (2-9) 2 ∂ ρ −iΩ1 ρ41 iΩ2 ρ23 iΩ ρ33 − ρ44 ∂t 43 iγ 42 γ 41 γ 31 γ 32 ρ43 ; (2-10) − i Δ2 − 2 with ρnm ρmn (m, n 1, 2, 3, 4 and m ≠ n) and ρ11 ρ22 ρ33 ρ44 1. γ mn represents decay rate from jmi to jni. According to [37–39], the relevant parameters are chosen as spin relaxation rate caused by phonon-assisted relaxation γ 21 0.1 μs−1 and the recombination of excitons rates γ 31 γ 42 0.025 ps−1 , which are about 0.016 meV in energy unit. It is obvious that γ 21 is several magnitudes smaller than γ 31 (γ 42 ). Beside, γ 32 and γ 41 are far smaller than γ 31 (γ 42 ), as is illustrated in [34]. As a result, we can neglect effects of γ 21 , γ 32 , and γ 41 on this system. As shown in Fig. 3, the probe field circulates in the unidirectional ring cavity while the driving field does not circle. With slowly varying envelope approximation, the dynamics of the probe field in an optical cavity are governed by Maxwell’s equation, which is given by
Fig. 3. Unidirectional ring cavity contains CNT quantum dots sample of length L. E I1 and E T1 are the incident and transmitted field, respectively. E 2 , B ∥ , and B ⊥ are the driving field, magnetic field component parallel to axis, and magnetic field component perpendicular to axis, respectively.
∂E 1 ∂E ν c 1 i 1 Pν1 : ∂t ∂z 2ε0
(3)
The c and ε0 are the light speed and permittivity of free space in Eq. (3), respectively. Pν1 denotes the induced polarization related with the transition j1i↔j3i and is determined by Pν1 N μ31 ρ31 in which N is the electron density of CNT quantum dots. Considering Eq. (3) in the steady-state condition, we acquire the field amplitude and matrix element related with the induced polarization as follows: ∂E 1 ν i 1 N μ31 ρ31 : ∂z 2ε0 c
(4)
For a perfect ring cavity, the incident field E I1 and the transmitted field E T1 meet the input–output relationship: pffiffiffiffi E 1 0 T E I1 RE 1 L; (5-1) ET E 1 L p1ffiffiffiffi ; T
(5-2)
where T and R are transmitted rate and reflected rate of the mirrors, and they meet R T 1, and L is the length of the CNT quantum dots sample. As exhibited in Fig. 3, mirror M 1 and mirror M 2 are both partially reflected and partially transmitted while all other mirrors are reflected.
Fig. 4. Input jyj versus output jxj for different values of spin–orbit coupling Δso . Blue dashed line is no spin–orbit coupling, which means Δso 0. Red solid line represents Δso 2.4 meV. Other parameters are Ω 0.5 meV, Ω2 0.01 meV, γ 31 γ 42 0.016 meV, Δ1 Δ2 0, and C 100 meV.
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Fig. 5. Input jyj versus output jxj for different values of the driving field Ω2 and detuning of driving field Δ2 . (a) Detuning of driving field is fixedly taken as Δ2 0. (b) Rabi frequency of driving field is fixedly taken as Ω2 0.5 meV. Other parameters are Ω 0.5 meV, γ 31 γ 42 0.016 meV, Δ1 0, Δso 2.4 meV, and C 100 meV.
μ E1 μ E1 By normalizing y ℏ31pffiffiffi and x 2ℏ31pffiffiffi and using the T T mean-field approximation [40], we can describe input–output formalism as follows: I
y 2x − iCρ31 x;
T
(6)
N ν1 Lμ231
where C 2ℏε0 cT is the cooperation parameter. Here ν1 , μ31 , and T are 2π × 2 × 1014 s−1 , 7.4 × 10−29 C · m [41], and 0.0165. Beside, the volume density of carbon nanotube is 1.7 × 1026 m−3 because the density of graphene is 5 × 1012 cm−2 [42]. As a result, the single CNT quantum dot of cooperation parameter C is about 100 meV. 3. RESULTS AND EXPLANATIONS In this section, we will investigate the effects of spin–orbit coupling, the intensity of the driving field, and the detunings of applied fields on OB. Under the steady-state condition, we solve Eq. (2) along with Eq. (6), and numerically obtain input–output relationships. We present a few numerical results, as shown in Figs. 4–6. First of all, we analyze the effect of spin–orbit coupling on OB.
Fig. 6. Input jyj versus output jxj for different values of detuning of probe field Δ1 . Other parameters are Ω 0.5 meV, Ω2 1 meV, γ 31 γ 42 0.016 meV, Δ2 2.4 meV, Δso 2.4 meV, and C 100 meV.
As exhibited in Fig. 4, we can see clearly that no OB occurs in the absence of spin–orbit coupling (blue dashed line) while OB arises in the presence of spin–orbit coupling (red solid line). The results of above Fig. 4 can be explained as follows. It is well known that EIT could not only reduce absorption but also enhance Kerr nonlinearity [43]. However, the perfect EIT cannot cause OB because the ideal EIT is transparent to the probe field. Thus, without spin–orbit coupling, the optical ring cavity is transparent for the probe field so that no OB occurs. In contrary, energy shift caused by spin–orbit coupling makes CNT opaque for probe field and strongly enhances Kerr nonlinearity [34], which is helpful to generate OB. Therefore, spin–orbit coupling is an essential factor for OB. In the following, we study how the intensity of the driving field and detuning of driving field affect OB. Figure 5(a) shows that the threshold of OB becomes large as the driving field increases. By applying an increasing driving field Ω2 , which couples with j4i and j2i, the absorption for probe field is enhanced. As a result, the threshold of OB becomes larger and the area of the hysteresis cycles wider. In addition, it is seen in Fig. 5(b) that the threshold of OB can decrease with the detuning of driving field increasing. Thus, the intensity of driving field and the detuning of driving field can control the threshold of OB. Figure 6 illustrates the role of detuning of probe field. It is remarkable that OM appears when detuning of the probe field is negative. The similar behavior of generating OM was also reported in the atomic system [7], where nonzero detunings of applied fields are an essential part. The blue solid line represents Δ1 0 and shows OB. However, when we choose the detuning of probe field Δ1 −1.2 meV and other parameters are fixed, we can observe OM, which is displayed by the red dashed line. It is because adjusting the detuning of probe filed can modify the coherence and induce higher order nonlinearity, which is important for the OM. Therefore, the y in Eq. (6) is not a cubic polynomial of the variable x in certain parameter regimes [44]. Thus, the detuning of the probe field can be used to manipulate the conversion from OB to OM with spin–orbit coupling.
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4. CONCLUSIONS In summary, we investigate the OB and OM behaviors in CNT with spin–orbit coupling. OB can be induced by the spin–orbit coupling and be manipulated efficiently by the intensity of the driving field as well as detuning of the driving field. Moreover, OM can also occur by choosing a negative detuning of probe field. Our results provide guidance for optimizing and controlling the optical switching process in CNT dots. Funding. Natural Science Foundation of Shaanxi Province (2014JM2-1001).
REFERENCES 1. H. M. Gibbs, Optical Bistability: Controlling Light With Light, Quantum Electronics—Principles and Applications (Academic, 1985). 2. A. Hurtado, M. Nami, I. D. Henning, M. J. Adams, and L. F. Lester, “Bistability patterns and nonlinear switching with very high contrast ratio in a 1550 nm quantum dash semiconductor laser,” Appl. Phys. Lett. 101, 161117 (2012). 3. G. Assanto, Z. Wang, D. J. Hagan, and E. W. Van Stryland, “All-optical modulation via nonlinear cascading in type II second-harmonic generation,” Appl. Phys. Lett. 67, 2120–2122 (1995). 4. H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesant, “Differential gain and bistability using a sodium-filled Fabry-Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976). 5. A. T. Rosenberger, L. A. Orozco, and H. J. Kimble, “Observation of absorptive bistability with two-level atoms in a ring cavity,” Phys. Rev. A 28, 2569–2572 (1983). 6. D. F. Wall and P. Zoller, “A coherent nonlinear mechanism for optical bistability from three level atoms,” Opt. Commun. 34, 260–264 (1980). 7. A. Joshi and M. Xiao, “Optical multistability in three-level atoms inside an optical ring cavity,” Phys. Rev. Lett. 91, 143904 (2003). 8. W. Harshawardhan and G. S. Agarwal, “Controlling optical bistability using electromagnetic-field-induced transparency and quantum interferences,” Phys. Rev. A 53, 1812–1817 (1996). 9. M. A. Antón, O. G. Calderón, and F. Carreño, “Optical bistability in V-type atoms driven by a coherent field in a broadband squeezed vacuum,” Phys. Lett. A 311, 297–312 (2003). 10. S. Singh, J. Rai, C. M. Bowden, and A. Postan, “Intrinsic optical bistability with squeezed vacuum,” Phys. Rev. A 45, 5160–5165 (1992). 11. P. Galatola, L. A. Lugiato, M. G. Porreca, and P. Tombesi, “Optical switching by variation of the squeezing vacuum,” Opt. Commun. 81, 175–178 (1991). 12. Z. Y. Chen, C. G. Du, S. Q. Gong, and Z. Z. Xu, “Optical bistability via squeezed vacuum input,” Phys. Lett. A 259, 15–19 (1999). 13. D. C. Cheng, C. P. Liu, and S. Q. Gong, “Optical bistability via amplitude and phase control of a microwave field,” Opt. Commun. 263, 111–115 (2006). 14. D. C. Cheng, C. P. Liu, and S. Q. Gong, “Optical bistability and multistability via the effect of spontaneously generated coherence in a three-level ladder-type atomic system,” Phys. Lett. A 332, 244–249 (2004). 15. X. D. Yang, C. Husko, C. W. Wong, M. B. Yu, and D. L. Kwong, “Observation of femtojoule optical bistability involving Fano resonances in high-Q/Vm silicon photonic crystal nanocavities,” Appl. Phys. Lett. 91, 051113 (2007). 16. J. H. Li, “Controllable optical bistability in a four-subband semiconductor quantum well system,” Phys. Rev. B 75, 155329 (2007). 17. M. A. Antón, F. Carreño, O. G. Calderón, and S. Melle, “Tunable alloptical bistability in a semiconductor quantum dot damped by a phase-dependent reservoir,” Opt. Commun. 281, 3301–3313 (2008). 18. Z. Wang, S. Zhen, X. Wu, J. Zhu, Z. Cao, and B. Yu, “Controllable optical bistability via tunneling induced transparency in quantum dot molecular,” Opt. Commun. 304, 7–10 (2013). 19. M. Kitano, T. Yabuzaki, and T. Ogawa, “Optical tristability,” Phys. Rev. Lett. 46, 926–929 (1981).
Research Article 20. P. R. Wallace, “The band theory of graphite,” Phys. Rev. 71, 622–634 (1947). 21. T. Ando, “Physics of graphene,” Progr. Theoret. Phys. Suppl. 176, 203–226 (2008). 22. E. Hendry, P. J. Hale, J. Moger, and A. K. Savchenko, “Coherent nonlinear optical response of graphene,” Phys. Rev. Lett. 105, 097401 (2010). 23. X. H. Yao and A. Belyanin, “Giant optical nonlinearity of graphene in a strong magnetic field,” Phys. Rev. Lett. 108, 255503 (2012). 24. N. M. R. Peres, Y. V. Bludov, J. E. Santos, A.-P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90, 125425 (2014). 25. D. Loss and D. P. DiVincenzo, “Quantum computation with quantum dots,” Phys. Rev. A 57, 120–126 (1998). 26. N. Mason, M. J. Biercuk, and C. M. Marcus, “Local gate control of a carbon nanotube double quantum dot,” Science 303, 655–658 (2004). 27. K. C. Nowack, F. H. L. Koppens, Y. V. Nazarov, and L. M. K. Vandersypen, “Coherent control of a single electron spin with electric fields,” Science 318, 1430–1433 (2007). 28. H. W. C. Postma, T. Teepen, Z. Yao, M. Grifoni, and C. Dekker, “Carbon nanotube single-electron transistors at room temperature,” Science 293, 76–79 (2001). 29. T. Ando, “Spin-orbit interaction in carbon nanotubes,” J. Phys. Soc. Jpn 69, 1757–1763 (2000). 30. D. Huertas-Hernando, F. Guinea, and A. Brataas, “Spin-orbit coupling in curved graphene, fullerenes, nanotubes, and nanotube caps,” Phys. Rev. B 74, 155426 (2006). 31. J. S. Jeong and H. W. Lee, “Curvature-enhanced spin-orbit coupling in a carbon nanotube,” Phys. Rev. B 80, 075409 (2009). 32. F. Kuemmeth, S. Hani, D. C. Ralph, and P. L. McEuen, “Coupling of spin and orbital motion of electrons in carbon nanotubes,” Nature 452, 448–452 (2008). 33. C. Galland and A. Imamoglu, “All-optical manipulation of electron spins in carbon-nanotube quantum dots,” Phys. Rev. Lett. 101, 157404 (2008). 34. H. Sun, X. L. Feng, S. Q. Gong, and C. H. Oh, “Giant cross-Kerr nonlinearity in carbon nanotube quantum dots with spin-orbit coupling,” Phys. Rev. B 79, 193404 (2009). 35. J. H. Pablo, J. Kong, H. S. J. van der Zant, C. Dekker, L. P. Kouwenhoven, and S. D. Franceschi, “Orbital Kondo effect in carbon nanotubes,” Nature 434, 484–488 (2005). 36. A. Makarovski, A. Zhukov, J. Liu, and G. Finkelstein, “SU(2) and SU(4) Kondo effects in carbon nanotube quantum dots,” Phys. Rev. B 75, 241407(R) (2007). 37. A. Hagen, M. Steiner, M. B. Raschke, C. Lienau, T. Hertel, H. H. Qian, A. J. Meixner, and A. Hartschuh, “Exponential decay lifetimes of excitons in individual single-walled carbon nanotubes,” Phys. Rev. Lett. 95, 197401 (2005). 38. K. M. Borysenko, Y. G. Semenov, K. W. Kim, and J. M. Zavada, “Electron spin relaxation via flexural phonon modes in semiconducting carbon nanotubes,” Phys. Rev. B 77, 205402 (2008). 39. S. Moriyama, T. Fuse, M. Suzuki, T. Yamaguchi, and K. Ishibashi, “Pulse-excited current measurements in carbon nanotube quantum dots,” J. Phys. 38, 9–12 (2006). 40. R. Bonifacio and L. A. Lugiato, “Optical bistability and cooperative effects in resonance fluorescence,” Phys. Rev. A 18, 1129–1144 (1978). 41. O. V. Ershova, Y. E. Lozovik, A. M. Popov, O. N. Bubel, E. F. Kislyakov, N. A. Poklonski, A. A. Knizhnik, and I. V. Lebedeva, “Control of the motion of nanoelectromechanical systems based on carbon nanotubes by electric fields,” J. Exp. Theor. Phys. 107, 653–661 (2008). 42. C. L. Ding, R. Yu, J. H. Li, X. Y. Hao, and Y. Wu, “Matched infrared soliton pairs in graphene under Landau quantization via four-wave mixing,” Phys. Rev. A 90, 043819 (2014). 43. S. E. Harris, J. E. Field, and A. Imamoğlu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990). 44. J. H. Li, X. Y. Lu, J. M. Luo, and Q. J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
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Vol. 55, No. 5 / February 10 2016 / Applied Optics
Optical bistability induced by spin–orbit coupling in the carbon-nanotube quantum dots WEI LIU, HONGJUN ZHANG,* HUI SUN, QIAOLIN ZHANG,
AND
DANDAN WANG
School of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, China *Corresponding author: [email protected] Received 30 October 2015; revised 3 January 2016; accepted 3 January 2016; posted 5 January 2016 (Doc. ID 253045); published 10 February 2016
We theoretically investigate steady-state behaviors of carbon nanotube quantum dots with spin–orbit coupling in a unidirectional ring cavity. Our results show that the spin–orbit coupling can induce optical bistability. In addition to the existence of spin–orbit coupling, switching optical bistability to optical multistability or vice versa is realized by adjusting detuning of the probe field. Our results show that the carbon nanotube quantum dots will possibly be a promising candidate for realizing all-optical switching. © 2016 Optical Society of America OCIS codes: (190.1450) Bistability; (190.3270) Kerr effect; (190.5970) Semiconductor nonlinear optics including MQW. http://dx.doi.org/10.1364/AO.55.001090
1. INTRODUCTION During the past several decades, optical bistability (OB) has attracted interest of many researchers due to its broad applications in all-optical switching [1], logic functions [2], all-optical transistor [3], and so on. OB usually occurs with a nonlinear medium placed in a unidirectional ring cavity [4–7], which constructs important feedback. It has been studied widely in atomic systems [8–14] and semiconducting schemes [15–18]. Harshawardhan and Agarwal [8] have reported that electromagnetically induced transparency (EIT) can lower the threshold of OB. The squeezed vacuum interacting with atoms [9–12] has been shown to have a remarkable effect on OB. The phase control has been used to manipulate the threshold of OB. For instance, in a closed system the threshold of OB has been controlled by the relative phase [13,17]. Particularly, Cheng et al. have found that the relative phase can also manipulate the threshold of OB with the existence of spontaneously generated coherence, although the system is not closed [14]. In addition, recent publications have shown that OB depends on Fano interference [15,16] and tunneling-induced transparency [18] in semiconducting schemes. Furthermore, optical multistability (OM) has been theoretically predicted [19] and experimentally carried out [7]. Recently, a single graphene layer has been given much attention for its zero-gap property in Dirac points [20], linear dispersion in the vicinity of Dirac points [21], and strong optical nonlinearity [22,23]. Peres et al. [24] have found that the inherent optical nonlinearity of graphene can lead to OB. Besides, some publications have predicted that graphene derivatives are potential candidates for spin-based applications [25–27] because of their long intrinsic coherence times and 1559-128X/16/051090-05$15/0$15.00 © 2016 Optical Society of America
the absence of nuclear spin [28], i.e., carbon nanotube (CNT). Some theories indicate that there is a strong spin–orbit coupling in CNT [29–31], although the spin–orbit coupling is weak in a single graphene [30]. Subsequently, it has been experimentally demonstrated that an electron’s orbit and its spin are strongly coupled [32]. The coupling supports parallel alignment of the orbital and spin magnetic moments for electrons and antiparallel alignment for holes. On the basis of the spin–orbit coupling, Galland and Imamoglu [33] have demonstrated theoretically all-optical manipulation of electron spins with a high fidelity. Moreover, the spin–orbit coupling, which leads to energy level shift, can effectively overcome group-velocity mismatching so that the giant cross-Kerr nonlinearity and coherent entanglement can be realized [34]. In this paper, motivated by spin–orbit coupling of CNT, we investigate the bistable behaviors of CNT quantum dots with spin–orbit coupling in a unidirectional ring cavity. We find that OB is induced by the spin–orbit coupling and can be manipulated by intensity and detuning of driving field. Beside, our results show OB is able to be switched to OM or vice versa with the spin–orbit coupling by adjusting the detuning of probe field. This paper is arranged as follows. The following section gives model and master equations, which describe the dynamic evolution of electrons with spin–orbit coupling; numerical results and explanations are presented in Section 3. Section 4 is the conclusion. 2. MODEL AND MASTER EQUATIONS As shown in Fig. 1(a), a quantum dot (QD) consisting of semiconducting CNT with a diameter d ∼ 1.2 nm is considered. The QD has a lowest optical transition in the near-infrared
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Fig. 1. (a) Nanotube with two top gates. The probe and driving fields are applied perpendicular to the CNT axis. (b) Energy diagram of the lowest electron and hole states in a nanotube quantum dot versus the applied axial magnetic field B ∥ .
(∼1500 nm). As shown in Fig. 1(b), the zero-field fourfold degeneracy [35,36] is split into two pairs of doublet states due to the existence of strong spin–orbit coupling. Two pairs of doublets, holding energy differences [33] Δeso ≈ 1.5 meV and Δhso ≈ 0.9 meV, have parallel and antiparallel spin and orbital magnetic moments. Here, SO represents spin-orbit coupling and eh is electron (hole). The doublet states can be separated by applying a magnetic field B ∥ parallel to the CNT axis because of Zeeman effects. To distinguish different states, we label the states with combinations of U D, eh, and arrows in Fig. 1(b) where U D signifies the positive (negative) orbital magnetic moment, eh represents electron (hole), and arrows stand for the projection of the spin along the CNT axis. Taken into account momentum conservation, the types of U → U and D → D are allowable optical transitions. In addition, the up and down electron (hole) spin states are coherently coupled by the magnetic field orthogonal to the CNT axis [34,35]. For convenience of calculating, Fig. 1(b) is converted to Fig. 2(a) equivalently. The probe field with angular frequency ν1 and driving field with angular frequency ν2 are applied to couple with j1i↔j3i and j2i↔j4i, respectively. At the same time, level j3i and level j4i are coupled by a transverse
magnetic field with angular frequency ν3, which is orthogonal to the CNT axis. Here, Δ1 ω31 − ν1 , Δ2 ω42 − ν2 , and Δ3 ω34 − ν3 are the detunings of probe field, driving field, and transverse magnetic field, respectively, in which ωij is the energy difference between jii and jji. For analyzing solely the effect of spin–orbit coupling, we introduce two parameters, Δ1 ω42 − ν1 and ΔSO ΔhSO ΔeSO , which is the energy shift caused by spin–orbit coupling. Then, Fig. 2(a) can be transformed into Fig. 2(b). Under electric-dipole and rotating wave approximations, the interaction Hamiltonian of Fig. 2(b) can be given as follows (ℏ 1): H int Δ1 Δso − Δ2 − Δ3 j2ih2j Δ1 Δso j3ih3j Δ1 Δso − Δ3 j4ih4j − Ω1 j1ih3j Ω2 j2ih4j Ωj4ih3j h:c:;
(1)
in which h.c. is Hermitian conjugation. Ω1 E 12ℏμ31 and Ω2 E 2 μ42 2ℏ are half of Rabi frequencies of probe field and driving field, where μnm denotes the dipole moment for the transition between jni and jmi. Magnetic Rabi frequency Ω is denoted by gμB B ⊥ 2ℏ , whereby Lande factor g ≈ 2 is for the electron and hole,
Fig. 2. (a) Simple diagram is applied to illustrate that CNT interacts with optical fields and magnetic field. (b) Adapted diagram for highlighting spin-orbit coupling.
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and μB is the Bohr magneton. Because of the coupling resonantly the transition between state j3i and state j4i, the detuning of the transverse magnetic field (Δ3 ) is zero. Consequently, the matrix equations are given as follows: ∂ ρ iΩ1 ρ31 − iΩ1 ρ13 γ 41 ρ44 γ 31 ρ33 ; (2-1) ∂t 11 ∂ ρ iΩ2 ρ42 − iΩ2 ρ24 γ 32 ρ33 γ 42 ρ44 ; (2-2) ∂t 22 ∂ ρ iΩ1 ρ13 − iΩ1 ρ31 iΩρ43 − iΩ ρ34 − γ 31 ρ33 ∂t 33 − γ 32 ρ33 ; (2-3) ∂ ρ −iΩ2 ρ42 iΩ2 ρ24 iΩ ρ34 − iΩρ43 ∂t 44 − γ 41 ρ44 − γ 42 ρ44 ; (2-4) ∂ iγ ρ21 −iΩ1 ρ23 iΩ2 ρ41 − i Δ1 Δso − Δ2 − 41 ρ21 ; 2 ∂t (2-5)
∂ ρ iΩ1 ρ11 − ρ33 iΩρ41 ∂t 31 iγ γ 32 ρ31 ; (2-6) − i Δ1 Δso − 31 2 ∂ ρ iΩ1 ρ12 iΩρ42 − iΩ2 ρ34 ∂t 32 iγ γ 32 γ 21 ρ32 ; (2-7) − i Δ1 Δso − 31 2 ∂ ρ −iΩ1 ρ43 iΩ2 ρ21 iΩρ31 ∂t 41 iγ γ 41 ρ41 ; − i Δ1 Δso − 42 (2-8) 2 ∂ ρ iΩ2 ρ22 − ρ44 iΩ ρ32 ∂t 42 iγ γ 41 γ 21 ρ42 ; − i Δ2 − 42 (2-9) 2 ∂ ρ −iΩ1 ρ41 iΩ2 ρ23 iΩ ρ33 − ρ44 ∂t 43 iγ 42 γ 41 γ 31 γ 32 ρ43 ; (2-10) − i Δ2 − 2 with ρnm ρmn (m, n 1, 2, 3, 4 and m ≠ n) and ρ11 ρ22 ρ33 ρ44 1. γ mn represents decay rate from jmi to jni. According to [37–39], the relevant parameters are chosen as spin relaxation rate caused by phonon-assisted relaxation γ 21 0.1 μs−1 and the recombination of excitons rates γ 31 γ 42 0.025 ps−1 , which are about 0.016 meV in energy unit. It is obvious that γ 21 is several magnitudes smaller than γ 31 (γ 42 ). Beside, γ 32 and γ 41 are far smaller than γ 31 (γ 42 ), as is illustrated in [34]. As a result, we can neglect effects of γ 21 , γ 32 , and γ 41 on this system. As shown in Fig. 3, the probe field circulates in the unidirectional ring cavity while the driving field does not circle. With slowly varying envelope approximation, the dynamics of the probe field in an optical cavity are governed by Maxwell’s equation, which is given by
Fig. 3. Unidirectional ring cavity contains CNT quantum dots sample of length L. E I1 and E T1 are the incident and transmitted field, respectively. E 2 , B ∥ , and B ⊥ are the driving field, magnetic field component parallel to axis, and magnetic field component perpendicular to axis, respectively.
∂E 1 ∂E ν c 1 i 1 Pν1 : ∂t ∂z 2ε0
(3)
The c and ε0 are the light speed and permittivity of free space in Eq. (3), respectively. Pν1 denotes the induced polarization related with the transition j1i↔j3i and is determined by Pν1 N μ31 ρ31 in which N is the electron density of CNT quantum dots. Considering Eq. (3) in the steady-state condition, we acquire the field amplitude and matrix element related with the induced polarization as follows: ∂E 1 ν i 1 N μ31 ρ31 : ∂z 2ε0 c
(4)
For a perfect ring cavity, the incident field E I1 and the transmitted field E T1 meet the input–output relationship: pffiffiffiffi E 1 0 T E I1 RE 1 L; (5-1) ET E 1 L p1ffiffiffiffi ; T
(5-2)
where T and R are transmitted rate and reflected rate of the mirrors, and they meet R T 1, and L is the length of the CNT quantum dots sample. As exhibited in Fig. 3, mirror M 1 and mirror M 2 are both partially reflected and partially transmitted while all other mirrors are reflected.
Fig. 4. Input jyj versus output jxj for different values of spin–orbit coupling Δso . Blue dashed line is no spin–orbit coupling, which means Δso 0. Red solid line represents Δso 2.4 meV. Other parameters are Ω 0.5 meV, Ω2 0.01 meV, γ 31 γ 42 0.016 meV, Δ1 Δ2 0, and C 100 meV.
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Fig. 5. Input jyj versus output jxj for different values of the driving field Ω2 and detuning of driving field Δ2 . (a) Detuning of driving field is fixedly taken as Δ2 0. (b) Rabi frequency of driving field is fixedly taken as Ω2 0.5 meV. Other parameters are Ω 0.5 meV, γ 31 γ 42 0.016 meV, Δ1 0, Δso 2.4 meV, and C 100 meV.
μ E1 μ E1 By normalizing y ℏ31pffiffiffi and x 2ℏ31pffiffiffi and using the T T mean-field approximation [40], we can describe input–output formalism as follows: I
y 2x − iCρ31 x;
T
(6)
N ν1 Lμ231
where C 2ℏε0 cT is the cooperation parameter. Here ν1 , μ31 , and T are 2π × 2 × 1014 s−1 , 7.4 × 10−29 C · m [41], and 0.0165. Beside, the volume density of carbon nanotube is 1.7 × 1026 m−3 because the density of graphene is 5 × 1012 cm−2 [42]. As a result, the single CNT quantum dot of cooperation parameter C is about 100 meV. 3. RESULTS AND EXPLANATIONS In this section, we will investigate the effects of spin–orbit coupling, the intensity of the driving field, and the detunings of applied fields on OB. Under the steady-state condition, we solve Eq. (2) along with Eq. (6), and numerically obtain input–output relationships. We present a few numerical results, as shown in Figs. 4–6. First of all, we analyze the effect of spin–orbit coupling on OB.
Fig. 6. Input jyj versus output jxj for different values of detuning of probe field Δ1 . Other parameters are Ω 0.5 meV, Ω2 1 meV, γ 31 γ 42 0.016 meV, Δ2 2.4 meV, Δso 2.4 meV, and C 100 meV.
As exhibited in Fig. 4, we can see clearly that no OB occurs in the absence of spin–orbit coupling (blue dashed line) while OB arises in the presence of spin–orbit coupling (red solid line). The results of above Fig. 4 can be explained as follows. It is well known that EIT could not only reduce absorption but also enhance Kerr nonlinearity [43]. However, the perfect EIT cannot cause OB because the ideal EIT is transparent to the probe field. Thus, without spin–orbit coupling, the optical ring cavity is transparent for the probe field so that no OB occurs. In contrary, energy shift caused by spin–orbit coupling makes CNT opaque for probe field and strongly enhances Kerr nonlinearity [34], which is helpful to generate OB. Therefore, spin–orbit coupling is an essential factor for OB. In the following, we study how the intensity of the driving field and detuning of driving field affect OB. Figure 5(a) shows that the threshold of OB becomes large as the driving field increases. By applying an increasing driving field Ω2 , which couples with j4i and j2i, the absorption for probe field is enhanced. As a result, the threshold of OB becomes larger and the area of the hysteresis cycles wider. In addition, it is seen in Fig. 5(b) that the threshold of OB can decrease with the detuning of driving field increasing. Thus, the intensity of driving field and the detuning of driving field can control the threshold of OB. Figure 6 illustrates the role of detuning of probe field. It is remarkable that OM appears when detuning of the probe field is negative. The similar behavior of generating OM was also reported in the atomic system [7], where nonzero detunings of applied fields are an essential part. The blue solid line represents Δ1 0 and shows OB. However, when we choose the detuning of probe field Δ1 −1.2 meV and other parameters are fixed, we can observe OM, which is displayed by the red dashed line. It is because adjusting the detuning of probe filed can modify the coherence and induce higher order nonlinearity, which is important for the OM. Therefore, the y in Eq. (6) is not a cubic polynomial of the variable x in certain parameter regimes [44]. Thus, the detuning of the probe field can be used to manipulate the conversion from OB to OM with spin–orbit coupling.
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4. CONCLUSIONS In summary, we investigate the OB and OM behaviors in CNT with spin–orbit coupling. OB can be induced by the spin–orbit coupling and be manipulated efficiently by the intensity of the driving field as well as detuning of the driving field. Moreover, OM can also occur by choosing a negative detuning of probe field. Our results provide guidance for optimizing and controlling the optical switching process in CNT dots. Funding. Natural Science Foundation of Shaanxi Province (2014JM2-1001).
REFERENCES 1. H. M. Gibbs, Optical Bistability: Controlling Light With Light, Quantum Electronics—Principles and Applications (Academic, 1985). 2. A. Hurtado, M. Nami, I. D. Henning, M. J. Adams, and L. F. Lester, “Bistability patterns and nonlinear switching with very high contrast ratio in a 1550 nm quantum dash semiconductor laser,” Appl. Phys. Lett. 101, 161117 (2012). 3. G. Assanto, Z. Wang, D. J. Hagan, and E. W. Van Stryland, “All-optical modulation via nonlinear cascading in type II second-harmonic generation,” Appl. Phys. Lett. 67, 2120–2122 (1995). 4. H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesant, “Differential gain and bistability using a sodium-filled Fabry-Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976). 5. A. T. Rosenberger, L. A. Orozco, and H. J. Kimble, “Observation of absorptive bistability with two-level atoms in a ring cavity,” Phys. Rev. A 28, 2569–2572 (1983). 6. D. F. Wall and P. Zoller, “A coherent nonlinear mechanism for optical bistability from three level atoms,” Opt. Commun. 34, 260–264 (1980). 7. A. Joshi and M. Xiao, “Optical multistability in three-level atoms inside an optical ring cavity,” Phys. Rev. Lett. 91, 143904 (2003). 8. W. Harshawardhan and G. S. Agarwal, “Controlling optical bistability using electromagnetic-field-induced transparency and quantum interferences,” Phys. Rev. A 53, 1812–1817 (1996). 9. M. A. Antón, O. G. Calderón, and F. Carreño, “Optical bistability in V-type atoms driven by a coherent field in a broadband squeezed vacuum,” Phys. Lett. A 311, 297–312 (2003). 10. S. Singh, J. Rai, C. M. Bowden, and A. Postan, “Intrinsic optical bistability with squeezed vacuum,” Phys. Rev. A 45, 5160–5165 (1992). 11. P. Galatola, L. A. Lugiato, M. G. Porreca, and P. Tombesi, “Optical switching by variation of the squeezing vacuum,” Opt. Commun. 81, 175–178 (1991). 12. Z. Y. Chen, C. G. Du, S. Q. Gong, and Z. Z. Xu, “Optical bistability via squeezed vacuum input,” Phys. Lett. A 259, 15–19 (1999). 13. D. C. Cheng, C. P. Liu, and S. Q. Gong, “Optical bistability via amplitude and phase control of a microwave field,” Opt. Commun. 263, 111–115 (2006). 14. D. C. Cheng, C. P. Liu, and S. Q. Gong, “Optical bistability and multistability via the effect of spontaneously generated coherence in a three-level ladder-type atomic system,” Phys. Lett. A 332, 244–249 (2004). 15. X. D. Yang, C. Husko, C. W. Wong, M. B. Yu, and D. L. Kwong, “Observation of femtojoule optical bistability involving Fano resonances in high-Q/Vm silicon photonic crystal nanocavities,” Appl. Phys. Lett. 91, 051113 (2007). 16. J. H. Li, “Controllable optical bistability in a four-subband semiconductor quantum well system,” Phys. Rev. B 75, 155329 (2007). 17. M. A. Antón, F. Carreño, O. G. Calderón, and S. Melle, “Tunable alloptical bistability in a semiconductor quantum dot damped by a phase-dependent reservoir,” Opt. Commun. 281, 3301–3313 (2008). 18. Z. Wang, S. Zhen, X. Wu, J. Zhu, Z. Cao, and B. Yu, “Controllable optical bistability via tunneling induced transparency in quantum dot molecular,” Opt. Commun. 304, 7–10 (2013). 19. M. Kitano, T. Yabuzaki, and T. Ogawa, “Optical tristability,” Phys. Rev. Lett. 46, 926–929 (1981).
Research Article 20. P. R. Wallace, “The band theory of graphite,” Phys. Rev. 71, 622–634 (1947). 21. T. Ando, “Physics of graphene,” Progr. Theoret. Phys. Suppl. 176, 203–226 (2008). 22. E. Hendry, P. J. Hale, J. Moger, and A. K. Savchenko, “Coherent nonlinear optical response of graphene,” Phys. Rev. Lett. 105, 097401 (2010). 23. X. H. Yao and A. Belyanin, “Giant optical nonlinearity of graphene in a strong magnetic field,” Phys. Rev. Lett. 108, 255503 (2012). 24. N. M. R. Peres, Y. V. Bludov, J. E. Santos, A.-P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90, 125425 (2014). 25. D. Loss and D. P. DiVincenzo, “Quantum computation with quantum dots,” Phys. Rev. A 57, 120–126 (1998). 26. N. Mason, M. J. Biercuk, and C. M. Marcus, “Local gate control of a carbon nanotube double quantum dot,” Science 303, 655–658 (2004). 27. K. C. Nowack, F. H. L. Koppens, Y. V. Nazarov, and L. M. K. Vandersypen, “Coherent control of a single electron spin with electric fields,” Science 318, 1430–1433 (2007). 28. H. W. C. Postma, T. Teepen, Z. Yao, M. Grifoni, and C. Dekker, “Carbon nanotube single-electron transistors at room temperature,” Science 293, 76–79 (2001). 29. T. Ando, “Spin-orbit interaction in carbon nanotubes,” J. Phys. Soc. Jpn 69, 1757–1763 (2000). 30. D. Huertas-Hernando, F. Guinea, and A. Brataas, “Spin-orbit coupling in curved graphene, fullerenes, nanotubes, and nanotube caps,” Phys. Rev. B 74, 155426 (2006). 31. J. S. Jeong and H. W. Lee, “Curvature-enhanced spin-orbit coupling in a carbon nanotube,” Phys. Rev. B 80, 075409 (2009). 32. F. Kuemmeth, S. Hani, D. C. Ralph, and P. L. McEuen, “Coupling of spin and orbital motion of electrons in carbon nanotubes,” Nature 452, 448–452 (2008). 33. C. Galland and A. Imamoglu, “All-optical manipulation of electron spins in carbon-nanotube quantum dots,” Phys. Rev. Lett. 101, 157404 (2008). 34. H. Sun, X. L. Feng, S. Q. Gong, and C. H. Oh, “Giant cross-Kerr nonlinearity in carbon nanotube quantum dots with spin-orbit coupling,” Phys. Rev. B 79, 193404 (2009). 35. J. H. Pablo, J. Kong, H. S. J. van der Zant, C. Dekker, L. P. Kouwenhoven, and S. D. Franceschi, “Orbital Kondo effect in carbon nanotubes,” Nature 434, 484–488 (2005). 36. A. Makarovski, A. Zhukov, J. Liu, and G. Finkelstein, “SU(2) and SU(4) Kondo effects in carbon nanotube quantum dots,” Phys. Rev. B 75, 241407(R) (2007). 37. A. Hagen, M. Steiner, M. B. Raschke, C. Lienau, T. Hertel, H. H. Qian, A. J. Meixner, and A. Hartschuh, “Exponential decay lifetimes of excitons in individual single-walled carbon nanotubes,” Phys. Rev. Lett. 95, 197401 (2005). 38. K. M. Borysenko, Y. G. Semenov, K. W. Kim, and J. M. Zavada, “Electron spin relaxation via flexural phonon modes in semiconducting carbon nanotubes,” Phys. Rev. B 77, 205402 (2008). 39. S. Moriyama, T. Fuse, M. Suzuki, T. Yamaguchi, and K. Ishibashi, “Pulse-excited current measurements in carbon nanotube quantum dots,” J. Phys. 38, 9–12 (2006). 40. R. Bonifacio and L. A. Lugiato, “Optical bistability and cooperative effects in resonance fluorescence,” Phys. Rev. A 18, 1129–1144 (1978). 41. O. V. Ershova, Y. E. Lozovik, A. M. Popov, O. N. Bubel, E. F. Kislyakov, N. A. Poklonski, A. A. Knizhnik, and I. V. Lebedeva, “Control of the motion of nanoelectromechanical systems based on carbon nanotubes by electric fields,” J. Exp. Theor. Phys. 107, 653–661 (2008). 42. C. L. Ding, R. Yu, J. H. Li, X. Y. Hao, and Y. Wu, “Matched infrared soliton pairs in graphene under Landau quantization via four-wave mixing,” Phys. Rev. A 90, 043819 (2014). 43. S. E. Harris, J. E. Field, and A. Imamoğlu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990). 44. J. H. Li, X. Y. Lu, J. M. Luo, and Q. J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
