Abstract

In this paper, a near-infrared (NIR) laser (with frequency ωNIR) exciting and an intense mid-infrared (MIR) bichromatic laser (with frequencies ω1 and ω2) driving scheme is proposed in semiconductors. It is predicted that if the frequency difference between the two components of the MIR laser, i.e. Ω = ω1-ω2, is in dozens of GHz frequency range, the emitting light contains spectra with frequencies ω = ωNIR + NΩ, where N is integer. In analogy with high-order THz sideband generation (HSG) in semiconductors, this phenomenon is named difference frequency sideband generation (DSG). Similar to the HSG case, the emitted sideband spectrum in DSG exhibits a nonperturbative plateau, where the intensity of the sideband remains approximately constant up to a cutoff frequency. The location of the cutoff frequency and its relationship with the frequency detuning of the NIR laser are discussed via the semi-classical saddle-point method.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Recently, the extensive study of the interaction of intense electromagnetic fields with semiconductors results in the reveal of many new effects, including dynamical Franz-Keldysh effect [1,2], non-linear excitonic effects [3,4], high-order harmonic generation [5–7], etc. The high-order THz sideband generation (HSG) is one of the most prominent effects [8–14], in which semiconductors are simultaneously illuminated by a weak near-infrared (NIR) laser with frequency ωNIR and an intense THz field with frequency ωTHz. The NIR laser excites electron–hole pairs, which will be accelerated by the intense THz field. The recombination of the generated electron–hole pairs will then produce sidebands with frequency evenly spaced by twice the THz frequency, i.e. ω = ωNIR + 2THz. HSG has received strong attentions thanks to its potential usefulness in many applications [9,15–19]. For example, since the sideband spectrum is a frequency comb, the HSG is expected to be useful in wide-band optical multiplexer. Moreover, considering the intense THz field as a controlling electric field, the HSG is a high speed electro-optical modulation process, which can be utilized to realize Tbit/s light modulation in optical communication.

In this work, we propose another driving scheme to obtain a cousin phenomenon of HSG, named difference frequency sideband generation (DSG), in which an intense mid-infrared (MIR) bichromatic laser with frequencies ω1 and ω2 (the frequency difference, i.e. Ω = ω1-ω2, is in dozens of GHz frequency range) is employed to replace the intense THz field. In this driving scheme, the electron–hole pairs created by the NIR laser are accelerated by the intense MIR laser, and the recombination of the electron–hole pairs emits sidebands with frequency evenly spaced by Ω, i.e. ω = ωNIR + NΩ. Figure 1 gives the schematic diagram of the frequency spectrum of the DSG. Similar to HSG, the generated sideband spectrum in DSG is also a frequency comb. However, the frequency spacing between the teeth of this comb isdifferent with that of the HSG. It enables the DSG’s capability to generate a new kind of frequency comb. Actually, we find that the DSG is more suitable for generating frequency comb with frequency spacing of dozens of GHz rather than THz, while the HSG is the opposite (more detail about this will be given in the following). This makes us think that the DSG is a new effect worthy of detailed study.

 

Fig. 1 The schematic diagram of the frequency spectrum of the DSG.

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Similar to the HSG, the emitted sideband spectrum in DSG exhibits a nonperturbative plateau, where the intensity of the sidebands remains approximately constant up to a cutoff frequency. The location of the cutoff frequency and its relationship with the frequency detuning of the NIR laser are discussed via semi-classical saddle-point method. We find that a positive detuning will induce equal frequency shift of the upper and lower cutoffs relative to the NIR laser frequency, thus the bandwidth of the sideband spectrum of the DSG is independent of the positive detuning. In the following, we begin our discussion.

2. Implement method of the driving scheme

In our driving scheme, the MIR bichromatic laser is one of the most important elements. Generally, a coherent MIR bichromatic laser can be written as

F(t)=F1eiθcos(ω1t)+F2cos(ω2t),
where F1 and F2 are the amplitudes of the two frequency components, and θ is their initial phase difference. If the laser is a continuous laser or a pulse with the pulse duration much larger than T=2π/(ω1ω2), the relative phase between the two frequency components will appear equivalently from 0 to 2π during the duration. In this case, there should be no influence of θ on the DSG. If the laser is a short pulse whose duration is about several T or shorter than T, the relative phase no longer appears equivalently from 0 to 2π, there must be influence of θ on the DSG, which is the so-called carrier envelope phase effect. In this paper, we consider only the former case, i.e. we do not pay our attention to the carrier envelope phase effect. Thus, without loss of generality we can assume θ=0.

If the two frequency components of the MIR bichromatic laser in Eq. (1) come from two independent laser sources, it is difficult to ensure the coherence of the two frequency components. To overcome this, we propose to generate the MIR bichromatic laser through a cosine amplitude modulation of a monochromatic MIR laser. Considering the monochromatic MIR laser is F0(t)=2Fcos(ωat) with ωa = (ω1 + ω2)/2 being the average frequency and the modulation frequency of the modulator is ωmod=Ω/2, the amplitude modulation and the obtained MIR bichromatic laser can be expressed as

F(t)=F0(t)cos(ωmodt)=2Fcos(ωat)cos(ωmodt)=Fcos(ω1t)+Fcos(ω2t).
In Eq. (2), the amplitudes of the two frequency components are the same, therefore in the following discussion we consider only the situation of F1 = F2 = F.

3. Comparison between DSG and HSG

As we propose to obtain the MIR bichromatic laser by modulating the amplitude of a monochromatic MIR laser beam, the frequency spacing of the comb spectrum of the DSG is determined by the modulation frequency of the modulator. Since the modulation frequency of current commercial electro-optical modulator is about 1~100GHz, the DSG is suitable for generating frequency comb with dozens of GHz frequency spacing. While in contrast, the HSG is not suitable for producing such kind of frequency comb when considering device integration. To illustrate this, we assume to generate a frequency comb with 100GHz frequency spacing using the HSG. Since the frequency spacing of the comb spectrum of the HSG is twice of the driving field (intense THz field) frequency, the frequency of the driving field should be chosen as 50GHz. The corresponding wavelength of this field is about 6 mm. Due to the limitation of diffraction limit, the focal spot obtained by focusing this field is about several cm in size, which is obviously not suitable for fabricating integrated optical devices. Therefore, to meet the current demand of high integration devices, the DSG is more suitable than the HSG for generating frequency comb with dozens of GHz frequency spacing.

4. Theories, numerical results, and discussions of DSG

In the above sections, some qualitative properties about the DSG have been discussed. In this section, we investigate the DSG quantitatively. To achieve this, we consider an effective mass model for a parabolic band material driven by the intense MIR bichromatic laser Eq. (2). The parabolic band approximation is reasonable under near resonance excitation and could significantly simplify our following calculation (leading to the integration over k in Eq. (6) can be performed analytically). The exciting NIR laser used is a continuous wave with the form of FNIR(t) = FNIRexp(−NIRt). In the single active electron approximation, the optical response of the semiconductor is determined by two-band (conduction band and valence band) Bloch equation

iρcv(k,t)t={12mcv[keA(t)]2+Egiγ2}ρcv(k,t)+dcvFNIR(t),
where ρcv = <c|ρ|ν> is the nondiagonal element of the density matrix with |c> and |v> respectively denoting the wave function of conduction band and valence band electrons, A(t) = -Fsin(ω1t)/ω1-Fsin(ω2t)/ω2 is the vector-potential of F(t), mcv is the reduced mass of electron-hole pair, Eg is the band gap of the semiconductor, γ2 is interband dephasing rate, and dcv is the interband transition dipole moment.

It should be noted that the excitation of electron–hole pair by the intense MIR laser is ignored in Eq. (3). This is reasonable since the recombination of the MIR laser excited electron–hole pair will emit high harmonics of the MIR laser frequency (high order harmonic generation) rather than the difference frequency sideband. In the meantime, there are of course influences of the MIR laser excitation on the DSG. It plays its role via exciting high carrier densities, increasing the carrier-carrier scattering rate, then leading to a high dephasing rate, which can drastically suppress the sideband generation. Thus, in the DSG, the intensity of the MIR laser should be appropriate. More details about this will be discussed in the following.

The Bloch Eq. (3) can be solved exactly. The expression of ρcv can be found to be

ρcv(k,t)=i0eiS(k,t,τ)/iωtγ2τdcvFNIRdτ,
with the semiclassical action
S(k,t,τ)=tτt[keA(t)]22mcvdt(ωNIREg)τ(ωωNIR)t,
where (ωωNIR) is the sideband energy relative to the NIR laser frequency and, τ denotes the delay between the recombination and the creation of the electron-hole pair. The interband polarization is given by the expectation value of the dipole moment operator. It reads
P(t)=1(2π)3dcvρcv(k,t)d3k
Here we consider that the working medium is bulk material and the integral over k is 3D integral. If the working medium is 2D material, such as semiconductor quantum well, the integral of k needs to be converted into 2D integral. Having Eq. (6), the sideband spectrum can be calculated via the Fourier transform of the polarization. Substituting Eqs. (4) and (5) into Eq. (6), then performing the integral over k, the polarization becomes
P(t)=idcvdcvFNIR(2π)302πmcviτ3ei(ωNIREg/)τiωNIRtγ2τeiκ(τ),×eiξ1(τ)cos(2ω1tω1τ)eiξ2(τ)cos(2ω2tω2τ)eiξa(τ)cos(2ωatωaτ)eiξΩ(τ)cos(ΩtΩτ/2)dτ
where κ(τ), ξ1(τ), ξ2(τ), ξa(τ) and ξΩ(τ) are introduced auxiliary functions, which are
κ(τ)=e2F22mcvτ[1cos(ω1τ)ω14+1cos(ω2τ)ω24]e2F2τ4mcv{1ω12+1ω22},
ξa(τ)=e2F2mcvω1ω2{12ωasin(ωaτ)2ω1ω2τsin(ω2τ2)sin(ω1τ2)},
ξ1(τ)=e2F22mcvω13{12sin(ω1τ)1ω1τ[1cos(ω1τ)]},
ξ2(τ)=e2F22mcvω23{12sin(ω2τ)1ω2τ[1cos(ω2τ)]},
ξΩ(τ)=e2F2mcvω1ω2{2ω1ω2τsin(ω2τ2)sin(ω1τ2)1Ωsin(Ωτ2)}.
According to the property of Bessel function eiβcosα=inJn(β)einα, it’s easy to obtain the Fourier transform of the interband polarization, which reads
P(ωNIR+Na2ωa+N12ω1+N22ω2+NΩΩ)=idcvdcvFNIR(2π)302πmcviτ3ei(ωNIREg/)τγ2τeiκ(τ)   ×iNaN1N2NΩJNa{ξa(τ)}JN1{ξ1(τ)}JN2{ξ2(τ)}JNΩ{ξΩ(τ)}ei(Naωa+N1ω1+N2ω2+NΩΩ/2)τdτ
The Eq. (9) is inspirational, in which NΩ determines the order of the difference frequency sidebands, while Na, N1 and N2 determine the order of high harmonic of frequencies ωa, ω1 and ω2, respectively. Since the cutoff frequency of the high harmonic generation is about 3.17Up with Up = e2F2/4mcvωa2 being the ponderomotive energy, appropriate electric field intensity F and frequencies of the MIR laser could ensure that the vast majority of the polarization of Eq. (9) comes from Na = N1 = N2 = 0, i.e. the relative intense of the harmonic is very low. This can be satisfied by choosing F = 8x102kV.cm−1 and ω1=301meV,ω2=300meV (in the following, we will use these parameters for calculating the DSG sideband intensity), which gives 3.17Up/ωa0.25 for GaAs crystal with mcv = 0.058me, where me is free electron mass. In this case, the polarization becomes
P(ωNIR+NΩΩ)=idcvdcvFNIR(2π)302πmcviτ3ei(ωNIREg/)τγ2τeiκ(τ)   ×iNΩJ0{ξa(τ)}J0{ξ1(τ)}J0{ξ2(τ)}JNΩ{ξΩ(τ)}eiNΩΩτ/2dτ
Until now, according to Eq. (10), a numerical calculation can be performed to explore the sideband intensity and spectrum shape. Since in our driving configuration, the MIR laser is intense, it may excite high carrier densities, leading to a high dephasing rate, which can drastically suppress the sideband generation (see the supplementary material of Ref [15].). So, to avoid the high dephasing, the electric field F used in our calculation should satisfy that the concentration of exciton created by the MIR laser is low. An appropriate F could be obtained through a simple estimate as follows. Firstly, for the case of exciting a two-level system by light with frequency ω, the optical susceptibility is χ1/(ωω0+iγ2) [20], where ω0 is the resonate frequency. Sinceω0=Eg in semiconductors, the linear optical absorption in semiconductors is αImχγ2/[(ωEg)2+(γ2)2]. We then consider that GaAs is the working material (whose band gap is Eg = 1.519eV) and choose that the NIR laser is near-resonant, the frequencies of the MIR laser are ω1~ω2~300meV, and the NIR laser intensity is ~105W·cm−2 (it is same as the NIR laser intensity of the HSG experiment in Ref [11], under which γ2 is about several meV and the sideband generation can occur). Under these conditions, we have ωNIREg~γ2~meV. Therefore, it can be calculated that the absorption to the NIR laser is about six orders of magnitude larger than that to the MIR laser, i.e. αNIR/αMIR~106. So, as long as the MIR laser intensity is no larger than six orders of magnitude of the NIR laser intensity, i.e. the MIR laser intensity is lower than 1011W·cm−2, the excitation of the MIR laser should be low and the majority of the excitons should be created by the NIR laser, which leads to the change of the dephasing rate induced by the MIR laser being ignorable. This can be satisfied by considering F = 8x102kV.cm−1, which results in the MIR laser intensity I0=4ncε0F22.24×1010Wcm2, where n = 3.3 is refractive index, ε0 is the vacuum permittivity and c is the speed of light in vacuum. With these knowledge, we calculate the sideband intensity by using mcv=0.058me, dcv=6.7eA, γ2=1meV, Eg=1.519eV, F=8×102kVcm-1, ω1=301meV, ω2=300meV (ωmod=Ω/2120GHzin this case) and ωNIREg=0. The results are shown in Fig. 2, where we have introducedχ(ωNIR+NΩΩ)=P(ωNIR+NΩΩ)/ε0FNIRto represent the sideband intensity.

 

Fig. 2 The intensity of the sideband spectrum. The black arrow gives the position of the cutoff frequency of the spectrum plateau, which is NmaxΩ=4Up.

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The numerical results in Fig. 2 clearly show that the DSG can occur in our MIR bichromatic laser driving scheme. We can see that the emitted sideband spectrum exhibits a plateau, where the intensity of the sidebands remains approximately constant up to a cutoff frequency. Similar to the HSG, the location of the cutoff frequency and its relationship with the frequency detuning of the NIR laser can be discussed via semi-classical saddle-point method. Firstly, by minimizing the action, we have saddle point equation

kS(k,t,τ)=0k=eτtτtA(t)dt.
The physical meaning of this equation is same as that interpreted in Ref [9]. It means that the electron must return the position of the hole to recombine. Substituting Eq. (11) into Eq. (5), the action becomes to be
S(t,τ)=κ(τ)ξΩ(τ)cos(ΩtΩτ/2)ξ1(τ)cos(2ω1tω1τ)              ξ2(τ)cos(2ω2tω2τ)ξ0(τ)cos(2ωatωaτ)Δτ              Na2ωatN12ω1tN22ω2tNΩΩt
where (ωωNIR)=Na2ωa+N12ω1+N22ω2+NΩΩ and Δ=ωNIREg are introduced. Equation (12) determines the phase of the sideband and, if we consider only the difference frequency sideband of the fundamental frequency ωNIR, the high frequency parts of Eq. (12) should be zero, i.e. Na = N1 = N2 = 0. Thus, the action becomes
S(t,τ)=κ(τ)ξΩ(τ)cos(ΩtΩτ/2)ΔτNΩΩt
In addition, to emit difference frequency sideband, the delay τ should be at about time scale of 1/Ω for the electron-hole pair to gain energy. With this knowledge, it can be estimated that e2F2[1cos(ωiτ)]/2mcvτωi42(Up/ωi)[1cos(ωiτ)]/ωiτ1×1031 under the same calculation conditions of Fig. 2 (Up/ωi0.08;ωiτωi/Ω300). Thus κ(τ) can be simplified as
κ(τ)e2F2τ4mcv{1ω12+1ω22}2Upτ.
Similarly, ξΩ(τ) can be simplified as
ξΩ(τ)e2F2mcvω1ω2Ωsin(Ωτ2)4UpΩsin(Ωτ2).
Substituting Eqs. (14) and (15) into Eq. (13), the action becomes
S(t,τ)=2Upτ+4UpΩsin(Ωτ2)cos(ΩtΩτ/2)ΔτNΩΩt.
Taking derivative of action (16) with respect to t and τ, respectively, and then minimizing the action again, we can obtain other two saddle point equations
τS(t,τ)=0cos(Ωτ2)cos(ΩtΩτ2)+sin(Ωτ2)sin(ΩtΩτ2)=Δ2Up2Up,
tS(t,τ)=0NΩΩ=4Upsin(Ωτ2)sin(ΩtΩτ/2)
Solving Eqs. (17a) and (17b) we have

NΩΩ=[1cos(Ωτ)](2UpΔ)±sin(Ωτ)4UpΔΔ2

According to Eq. (18), the dimensionless difference frequency sideband frequency shift NΩΩ/Up as a function of the delay time could be calculated numerically. The results are shown in Fig. 3 for various frequency detuning △. The results present that the cutoffs of the DSG plateau depend on the frequency detuning. When △ = 0, the upper cutoff frequency of the spectrum plateau is 4Up and, the lower cutoff frequency is 0. This agrees with the result of Fig. 2. When △≠0, a positive detuning will induce equal shift of the upper and lower cutoffs, and the shift equals to △. To demonstrate this, we calculate the intensity of the difference frequency sideband spectrum under △ = 0.5Up, Up and 1.5Up with same calculation conditions as Fig. 2. The numerical results are shown in Fig. 4(a). Here we declare that in order to keep the sideband spectrum of the three cases from overlapping each other so that the spectrum cutoffs can be displayed clearly, the sideband spectrum of △ = Up has been shifted up by 1 and the sideband spectrum of △ = 1.5Up has been shifted down by 1 in Fig. 4(a). We can see that the cutoffs of the calculated sideband spectrum in Fig. 4(a) agree well with the results of Figs. 3(b)-3(d). Moreover, it can be found that the bandwidth of the sideband spectrum is independent of the positive detuning and, the frequency range of the plateau is always from Eg to Eg + 4Up. This can be seen clearly from Fig. 4(b), in which the color indicates the sideband intensity and, the red and black dotted line give the upper and lower cutoffs extracted through Eq. (18), respectively.

 

Fig. 3 The dimensionless sideband frequency shift NΩΩ/Up as a function of the delay time for various frequency detuning △. Where (a) △ = 0; (b) △ = 0.5Up; (c) △ = Up and (d) △ = 1.5Up. The red and blue curves are the results of NΩΩ=[1cos(Ωτ)](2UpΔ)+sin(Ωτ)4UpΔΔ2 and NΩΩ=[1cos(Ωτ)](2UpΔ)sin(Ωτ)4UpΔΔ2 respectively, which coincide with each other in (a).

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Fig. 4 (a) The intensity of the sideband spectrum under △ = 0.5Up, Up and 1.5Up. (b) The intensity of the sideband spectrum as a function of △ and the sideband frequency. The black and red dotted lines in (b) are respective the upper and lower cutoffs extracted through Eq. (15). The calculation conditions are same as that of Fig. 2 except △.

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5. Conclusion

In conclusion, DSG is predicted when a semiconductor is excited by a NIR laser and driven by an intense MIR bichromatic laser at same time. The sideband spectrum of the DSG is a frequency comb which exhibits a nonperturbative plateau. The location of the cutoff frequency of the sideband plateau is discussed via semi-classical saddle-point method. This work provides a deeper insight in the aspect of the interaction of intense electromagnetic fields with semiconductors.

Funding

National Natural Science Foundations of China (NSFC) (61535004, 61735009, 61675052, 61705050, 61765004, 61640409, 11604050), Guangxi project (AD17195074), National Defense Foundation of China (6140414030102), National Key R&D Program of China (2017YFB0405501), Guangxi Natural Science Foundation of China (2017GXNSFAA198048, 2016GXNSFAA380006), Guangxi Project for ability enhancement of young and middle-aged university teacher (2018KY0200).

References

1. K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998). [CrossRef]  

2. K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010). [CrossRef]  

3. B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012). [CrossRef]  

4. S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005). [CrossRef]   [PubMed]  

5. G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014). [CrossRef]   [PubMed]  

6. S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011). [CrossRef]  

7. T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017). [CrossRef]  

8. R.-B. Liu and B.-F. Zhu, “High-order THz-sideband generation in semiconductors,” AIP Conf. Proc. 893, 1455–1456 (2007). [CrossRef]  

9. J.-Y. Yan, “Theory of excitonic high-order sideband generation in semiconductors under a strong terahertz field,” Phys. Rev. B Condens. Matter Mater. Phys. 78(7), 075204 (2008). [CrossRef]  

10. J. A. Crosse and R.-B. Liu, “Quantum-coherence-induced second plateau in high-sideband generation,” Phys. Rev. B Condens. Matter Mater. Phys. 89(12), 121202 (2014). [CrossRef]  

11. B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012). [CrossRef]   [PubMed]  

12. H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013). [CrossRef]   [PubMed]  

13. H. Liu and W. She, “Theory of effect of near-infrared laser polarization direction on high-order terahertz sideband generation in semiconductors,” Opt. Express 23(8), 10680–10686 (2015). [CrossRef]   [PubMed]  

14. H. Liu and X. Zhang, “Theory of controlling band-width broadening in terahertz sideband generation in semiconductors by a direct current electric field,” Opt. Commun. 387, 37–42 (2017). [CrossRef]  

15. J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5(1), 4854 (2014). [CrossRef]   [PubMed]  

16. B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013). [CrossRef]  

17. F. Yang and R.-B. Liu, “Berry phases of quantum trajectories of optically excited electron–hole pairs in semiconductors under strong terahertz fields,” New J. Phys. 15(11), 115005 (2013). [CrossRef]  

18. F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014). [CrossRef]  

19. J.-Y. Yan, “High-order sideband generation in a semiconductor quantum well driven by two orthogonal terahertz fields,” J. Appl. Phys. 122(8), 084306 (2017). [CrossRef]  

20. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th ed. (World Scientific Publishing Company, 2009).

References

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  1. K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
    [Crossref]
  2. K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010).
    [Crossref]
  3. B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
    [Crossref]
  4. S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
    [Crossref] [PubMed]
  5. G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
    [Crossref] [PubMed]
  6. S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
    [Crossref]
  7. T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017).
    [Crossref]
  8. R.-B. Liu and B.-F. Zhu, “High-order THz-sideband generation in semiconductors,” AIP Conf. Proc. 893, 1455–1456 (2007).
    [Crossref]
  9. J.-Y. Yan, “Theory of excitonic high-order sideband generation in semiconductors under a strong terahertz field,” Phys. Rev. B Condens. Matter Mater. Phys. 78(7), 075204 (2008).
    [Crossref]
  10. J. A. Crosse and R.-B. Liu, “Quantum-coherence-induced second plateau in high-sideband generation,” Phys. Rev. B Condens. Matter Mater. Phys. 89(12), 121202 (2014).
    [Crossref]
  11. B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012).
    [Crossref] [PubMed]
  12. H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
    [Crossref] [PubMed]
  13. H. Liu and W. She, “Theory of effect of near-infrared laser polarization direction on high-order terahertz sideband generation in semiconductors,” Opt. Express 23(8), 10680–10686 (2015).
    [Crossref] [PubMed]
  14. H. Liu and X. Zhang, “Theory of controlling band-width broadening in terahertz sideband generation in semiconductors by a direct current electric field,” Opt. Commun. 387, 37–42 (2017).
    [Crossref]
  15. J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5(1), 4854 (2014).
    [Crossref] [PubMed]
  16. B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013).
    [Crossref]
  17. F. Yang and R.-B. Liu, “Berry phases of quantum trajectories of optically excited electron–hole pairs in semiconductors under strong terahertz fields,” New J. Phys. 15(11), 115005 (2013).
    [Crossref]
  18. F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014).
    [Crossref]
  19. J.-Y. Yan, “High-order sideband generation in a semiconductor quantum well driven by two orthogonal terahertz fields,” J. Appl. Phys. 122(8), 084306 (2017).
    [Crossref]
  20. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th ed. (World Scientific Publishing Company, 2009).

2017 (3)

T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017).
[Crossref]

H. Liu and X. Zhang, “Theory of controlling band-width broadening in terahertz sideband generation in semiconductors by a direct current electric field,” Opt. Commun. 387, 37–42 (2017).
[Crossref]

J.-Y. Yan, “High-order sideband generation in a semiconductor quantum well driven by two orthogonal terahertz fields,” J. Appl. Phys. 122(8), 084306 (2017).
[Crossref]

2015 (1)

2014 (4)

F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014).
[Crossref]

J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5(1), 4854 (2014).
[Crossref] [PubMed]

J. A. Crosse and R.-B. Liu, “Quantum-coherence-induced second plateau in high-sideband generation,” Phys. Rev. B Condens. Matter Mater. Phys. 89(12), 121202 (2014).
[Crossref]

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

2013 (3)

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013).
[Crossref]

F. Yang and R.-B. Liu, “Berry phases of quantum trajectories of optically excited electron–hole pairs in semiconductors under strong terahertz fields,” New J. Phys. 15(11), 115005 (2013).
[Crossref]

2012 (2)

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012).
[Crossref] [PubMed]

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

2011 (1)

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

2010 (1)

K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010).
[Crossref]

2008 (1)

J.-Y. Yan, “Theory of excitonic high-order sideband generation in semiconductors under a strong terahertz field,” Phys. Rev. B Condens. Matter Mater. Phys. 78(7), 075204 (2008).
[Crossref]

2007 (1)

R.-B. Liu and B.-F. Zhu, “High-order THz-sideband generation in semiconductors,” AIP Conf. Proc. 893, 1455–1456 (2007).
[Crossref]

2005 (1)

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[Crossref] [PubMed]

1998 (1)

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Agostini, P.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

Akiyama, H.

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Allen, S. J.

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Banks, H.

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013).
[Crossref]

Birkedal, V.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[Crossref] [PubMed]

Birnir, B.

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Brabec, T.

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

Carter, S. G.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[Crossref] [PubMed]

Chatterjee, S.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Citrin, D. S.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[Crossref] [PubMed]

Coldren, L. A.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[Crossref] [PubMed]

Corkum, P. B.

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

Crosse, J. A.

J. A. Crosse and R.-B. Liu, “Quantum-coherence-induced second plateau in high-sideband generation,” Phys. Rev. B Condens. Matter Mater. Phys. 89(12), 121202 (2014).
[Crossref]

J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5(1), 4854 (2014).
[Crossref] [PubMed]

DiChiara, A. D.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

DiMauro, L. F.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

Ewers, B.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Ghimire, S.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

Gibbs, H. M.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Gossard, A. C.

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

Hirori, H.

K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010).
[Crossref]

Huang, T.

T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017).
[Crossref]

Jauho, A.-P.

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Johnsen, K.

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Kadoya, Y.

K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010).
[Crossref]

Khitrova, G.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Kira, M.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Klettke, A. C.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Klug, D. D.

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

Koch, M.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Koch, S. W.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Kono, J.

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Koster, N. S.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Lan, P.

T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017).
[Crossref]

Li, L.

T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017).
[Crossref]

Liu, H.

H. Liu and X. Zhang, “Theory of controlling band-width broadening in terahertz sideband generation in semiconductors by a direct current electric field,” Opt. Commun. 387, 37–42 (2017).
[Crossref]

H. Liu and W. She, “Theory of effect of near-infrared laser polarization direction on high-order terahertz sideband generation in semiconductors,” Opt. Express 23(8), 10680–10686 (2015).
[Crossref] [PubMed]

Liu, R.

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

Liu, R. B.

J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5(1), 4854 (2014).
[Crossref] [PubMed]

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012).
[Crossref] [PubMed]

Liu, R.-B.

J. A. Crosse and R.-B. Liu, “Quantum-coherence-induced second plateau in high-sideband generation,” Phys. Rev. B Condens. Matter Mater. Phys. 89(12), 121202 (2014).
[Crossref]

F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014).
[Crossref]

F. Yang and R.-B. Liu, “Berry phases of quantum trajectories of optically excited electron–hole pairs in semiconductors under strong terahertz fields,” New J. Phys. 15(11), 115005 (2013).
[Crossref]

R.-B. Liu and B.-F. Zhu, “High-order THz-sideband generation in semiconductors,” AIP Conf. Proc. 893, 1455–1456 (2007).
[Crossref]

Liu, X.

T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017).
[Crossref]

Lu, P.

T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017).
[Crossref]

Mack, S.

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

Maslov, A. V.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[Crossref] [PubMed]

McDonald, C. R.

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

Nagai, M.

K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010).
[Crossref]

Noda, T.

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Nordstrom, K. B.

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Orlando, G.

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

Reis, D. A.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

Sakaki, H.

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Satoh, N.

K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010).
[Crossref]

She, W.

Sherwin, M. S.

J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5(1), 4854 (2014).
[Crossref] [PubMed]

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013).
[Crossref]

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012).
[Crossref] [PubMed]

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[Crossref] [PubMed]

Shinokita, K.

K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010).
[Crossref]

Sistrunk, E.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

Tanaka, K.

K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010).
[Crossref]

Vampa, G.

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

Wang, C. S.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[Crossref] [PubMed]

Woscholski, R.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Xu, X.

J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5(1), 4854 (2014).
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F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014).
[Crossref]

Yan, J.-Y.

J.-Y. Yan, “High-order sideband generation in a semiconductor quantum well driven by two orthogonal terahertz fields,” J. Appl. Phys. 122(8), 084306 (2017).
[Crossref]

J.-Y. Yan, “Theory of excitonic high-order sideband generation in semiconductors under a strong terahertz field,” Phys. Rev. B Condens. Matter Mater. Phys. 78(7), 075204 (2008).
[Crossref]

Yang, F.

F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014).
[Crossref]

F. Yang and R.-B. Liu, “Berry phases of quantum trajectories of optically excited electron–hole pairs in semiconductors under strong terahertz fields,” New J. Phys. 15(11), 115005 (2013).
[Crossref]

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

Zaks, B.

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013).
[Crossref]

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012).
[Crossref] [PubMed]

Zhang, X.

H. Liu and X. Zhang, “Theory of controlling band-width broadening in terahertz sideband generation in semiconductors by a direct current electric field,” Opt. Commun. 387, 37–42 (2017).
[Crossref]

Zhu, B.-F.

R.-B. Liu and B.-F. Zhu, “High-order THz-sideband generation in semiconductors,” AIP Conf. Proc. 893, 1455–1456 (2007).
[Crossref]

Zhu, X.

T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017).
[Crossref]

AIP Conf. Proc. (1)

R.-B. Liu and B.-F. Zhu, “High-order THz-sideband generation in semiconductors,” AIP Conf. Proc. 893, 1455–1456 (2007).
[Crossref]

Appl. Phys. Lett. (2)

K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010).
[Crossref]

B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013).
[Crossref]

J. Appl. Phys. (1)

J.-Y. Yan, “High-order sideband generation in a semiconductor quantum well driven by two orthogonal terahertz fields,” J. Appl. Phys. 122(8), 084306 (2017).
[Crossref]

Nat. Commun. (1)

J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5(1), 4854 (2014).
[Crossref] [PubMed]

Nat. Phys. (1)

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

Nature (1)

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012).
[Crossref] [PubMed]

New J. Phys. (2)

F. Yang and R.-B. Liu, “Berry phases of quantum trajectories of optically excited electron–hole pairs in semiconductors under strong terahertz fields,” New J. Phys. 15(11), 115005 (2013).
[Crossref]

F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014).
[Crossref]

Opt. Commun. (1)

H. Liu and X. Zhang, “Theory of controlling band-width broadening in terahertz sideband generation in semiconductors by a direct current electric field,” Opt. Commun. 387, 37–42 (2017).
[Crossref]

Opt. Express (1)

Phys. Rev. A (Coll. Park) (1)

T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017).
[Crossref]

Phys. Rev. B Condens. Matter Mater. Phys. (3)

J.-Y. Yan, “Theory of excitonic high-order sideband generation in semiconductors under a strong terahertz field,” Phys. Rev. B Condens. Matter Mater. Phys. 78(7), 075204 (2008).
[Crossref]

J. A. Crosse and R.-B. Liu, “Quantum-coherence-induced second plateau in high-sideband generation,” Phys. Rev. B Condens. Matter Mater. Phys. 89(12), 121202 (2014).
[Crossref]

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Phys. Rev. Lett. (3)

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
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G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

Science (1)

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
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Other (1)

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th ed. (World Scientific Publishing Company, 2009).

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Figures (4)

Fig. 1
Fig. 1 The schematic diagram of the frequency spectrum of the DSG.
Fig. 2
Fig. 2 The intensity of the sideband spectrum. The black arrow gives the position of the cutoff frequency of the spectrum plateau, which is N max Ω = 4 U p .
Fig. 3
Fig. 3 The dimensionless sideband frequency shift N Ω Ω / U p as a function of the delay time for various frequency detuning △. Where (a) △ = 0; (b) △ = 0.5Up; (c) △ = Up and (d) △ = 1.5Up. The red and blue curves are the results of N Ω Ω = [ 1 cos ( Ω τ ) ] ( 2 U p Δ ) + sin ( Ω τ ) 4 U p Δ Δ 2 and N Ω Ω = [ 1 cos ( Ω τ ) ] ( 2 U p Δ ) sin ( Ω τ ) 4 U p Δ Δ 2 respectively, which coincide with each other in (a).
Fig. 4
Fig. 4 (a) The intensity of the sideband spectrum under △ = 0.5Up, Up and 1.5Up. (b) The intensity of the sideband spectrum as a function of △ and the sideband frequency. The black and red dotted lines in (b) are respective the upper and lower cutoffs extracted through Eq. (15). The calculation conditions are same as that of Fig. 2 except △.

Equations (23)

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F ( t ) = F 1 e i θ cos ( ω 1 t ) + F 2 cos ( ω 2 t ) ,
F ( t ) = F 0 ( t ) cos ( ω mod t ) = 2 F cos ( ω a t ) cos ( ω mod t ) = F cos ( ω 1 t ) + F cos ( ω 2 t ) .
i ρ c v ( k , t ) t = { 1 2 m c v [ k e A ( t ) ] 2 + E g i γ 2 } ρ c v ( k , t ) + d c v F NIR ( t ) ,
ρ c v ( k , t ) = i 0 e i S ( k , t , τ ) / i ω t γ 2 τ d c v F NIR d τ ,
S ( k , t , τ ) = t τ t [ k e A ( t ) ] 2 2 m c v d t ( ω NIR E g ) τ ( ω ω NIR ) t ,
P ( t ) = 1 ( 2 π ) 3 d c v ρ c v ( k , t ) d 3 k
P ( t ) = i d c v d c v F NIR ( 2 π ) 3 0 2 π m c v i τ 3 e i ( ω NIR E g / ) τ i ω NIR t γ 2 τ e i κ ( τ ) , × e i ξ 1 ( τ ) cos ( 2 ω 1 t ω 1 τ ) e i ξ 2 ( τ ) cos ( 2 ω 2 t ω 2 τ ) e i ξ a ( τ ) cos ( 2 ω a t ω a τ ) e i ξ Ω ( τ ) cos ( Ω t Ω τ / 2 ) d τ
κ ( τ ) = e 2 F 2 2 m c v τ [ 1 cos ( ω 1 τ ) ω 1 4 + 1 cos ( ω 2 τ ) ω 2 4 ] e 2 F 2 τ 4 m c v { 1 ω 1 2 + 1 ω 2 2 } ,
ξ a ( τ ) = e 2 F 2 m c v ω 1 ω 2 { 1 2 ω a sin ( ω a τ ) 2 ω 1 ω 2 τ sin ( ω 2 τ 2 ) sin ( ω 1 τ 2 ) } ,
ξ 1 ( τ ) = e 2 F 2 2 m c v ω 1 3 { 1 2 sin ( ω 1 τ ) 1 ω 1 τ [ 1 cos ( ω 1 τ ) ] } ,
ξ 2 ( τ ) = e 2 F 2 2 m c v ω 2 3 { 1 2 sin ( ω 2 τ ) 1 ω 2 τ [ 1 cos ( ω 2 τ ) ] } ,
ξ Ω ( τ ) = e 2 F 2 m c v ω 1 ω 2 { 2 ω 1 ω 2 τ sin ( ω 2 τ 2 ) sin ( ω 1 τ 2 ) 1 Ω sin ( Ω τ 2 ) } .
P ( ω NIR + N a 2 ω a + N 1 2 ω 1 + N 2 2 ω 2 + N Ω Ω ) = i d c v d c v F NIR ( 2 π ) 3 0 2 π m c v i τ 3 e i ( ω NIR E g / ) τ γ 2 τ e i κ ( τ )       × i N a N 1 N 2 N Ω J N a { ξ a ( τ ) } J N 1 { ξ 1 ( τ ) } J N 2 { ξ 2 ( τ ) } J N Ω { ξ Ω ( τ ) } e i ( N a ω a + N 1 ω 1 + N 2 ω 2 + N Ω Ω / 2 ) τ d τ
P ( ω NIR + N Ω Ω ) = i d c v d c v F NIR ( 2 π ) 3 0 2 π m c v i τ 3 e i ( ω NIR E g / ) τ γ 2 τ e i κ ( τ )       × i N Ω J 0 { ξ a ( τ ) } J 0 { ξ 1 ( τ ) } J 0 { ξ 2 ( τ ) } J N Ω { ξ Ω ( τ ) } e i N Ω Ω τ / 2 d τ
k S ( k , t , τ ) = 0 k = e τ t τ t A ( t ) d t .
S ( t , τ ) = κ ( τ ) ξ Ω ( τ ) cos ( Ω t Ω τ / 2 ) ξ 1 ( τ ) cos ( 2 ω 1 t ω 1 τ )                             ξ 2 ( τ ) cos ( 2 ω 2 t ω 2 τ ) ξ 0 ( τ ) cos ( 2 ω a t ω a τ ) Δ τ                             N a 2 ω a t N 1 2 ω 1 t N 2 2 ω 2 t N Ω Ω t
S ( t , τ ) = κ ( τ ) ξ Ω ( τ ) cos ( Ω t Ω τ / 2 ) Δ τ N Ω Ω t
κ ( τ ) e 2 F 2 τ 4 m c v { 1 ω 1 2 + 1 ω 2 2 } 2 U p τ .
ξ Ω ( τ ) e 2 F 2 m c v ω 1 ω 2 Ω sin ( Ω τ 2 ) 4 U p Ω sin ( Ω τ 2 ) .
S ( t , τ ) = 2 U p τ + 4 U p Ω sin ( Ω τ 2 ) cos ( Ω t Ω τ / 2 ) Δ τ N Ω Ω t .
τ S ( t , τ ) = 0 cos ( Ω τ 2 ) cos ( Ω t Ω τ 2 ) + sin ( Ω τ 2 ) sin ( Ω t Ω τ 2 ) = Δ 2 U p 2 U p ,
t S ( t , τ ) = 0 N Ω Ω = 4 U p sin ( Ω τ 2 ) sin ( Ω t Ω τ / 2 )
N Ω Ω = [ 1 cos ( Ω τ ) ] ( 2 U p Δ ) ± sin ( Ω τ ) 4 U p Δ Δ 2

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