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Abstract
Dynamic localization has been thoroughly studied since 1986 by Dunlap in superlattice structures. However, its implications for terahertz (THz) radiation have not been fully explored. Here, we investigate the interplay between dynamic localization and THz radiation generation in semiconductor structures. Utilizing a two-color laser field, we reveal that intraband current is the primary source of THz radiation. Furthermore, we identify minima in THz radiation yield at specific laser field strengths, indicating the presence of dynamic localization. The relative phase of the two-color laser field and dephasing time can manipulate the extent and range of dynamic localization, thereby influencing the efficiency of THz radiation. Our findings provide valuable insights into simultaneous investigations on materials across different time scales.
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1. Introduction
In the past decade, extensive research has been dedicated to strong field emission from semiconductors, sparked by Ghimire’s observation of high harmonics generation (HHG) from ZnO using a 3250 nm laser field in 2010 [1]. This pioneering achievement opened up a new frontier in the field of strong field physics in solids. Focusing on identifying the mechanism, connection, and differentiation between the atomic gas cases are often being made with fruitful insights [2–9]. HHG is pivotal for attosecond pulse generation [10–12], providing insights into the ultrafast electronic dynamics of various systems, including atoms, molecules, plasmas, and solids. Utilizing a few-cycle laser field enhances the damage threshold of semiconductors, enabling the generation of a high harmonic spectrum spanning from below the minimum bandgap to higher bandgap energies.
In recent years, both experimental and theoretical studies have surged [4–7], seeking to understand the mechanisms behind HHG in solids. They conclude that the harmonic spectrum results from two distinct electric dynamics: intraband electronic nonlinear motion and interband polarization involving coupled electrons and holes [13,14], as shown in Fig. 1. When subjected to a far-infrared laser field, electrons residing in the highest valence band (VB) tunnel ionize into the conduction band (CB), resulting in the creation of electron-hole pairs. These pairs are subsequently accelerated within the CB and VB, exhibiting nonlinear responses to the laser field governed by the dispersion relations. The nonlinear motion of electrons, including Bloch oscillations (BO) [15–17] and Wannier-Stark ladder (WSL) states [18–20], induces a nonlinear current and generates lower frequency nonlinear emission, known as the intraband mechanism [14]. Upon the reversal of the laser field, electrons recombine with the previously created holes in the VB due to the polarization between the CB and VB. This process gives rise to harmonic spectrum with higher frequencies, with the cut-off energy determined by the band gap at the corresponding instantaneous crystal momentum. This mechanism is referred to as the interband mechanism [13]. Through the interplay of these intricate dynamics, HHG in solids offers a rich and complex phenomenon that requires further exploration and understanding.
Fig. 1. Electronic dynamics schematic under strong laser fields in a typical band strucure. The process of HHG is described by the purple lines, including intraband nonlinear motion induced radiation and interband polarization induced emission. The blue lines represent the process of THz radiation based on intraband nonlinear motion of electrons. The yellow lines represent the Bloch oscillations of electrons within the energy band. When the frequency of Bloch oscillations resonates with the frequency of femtosecond driving light, dynamic localization of electrons occurs.
Ultrastrong and ultrafast laser pulses can also drive down-conversion of frequency, resulting in the generation of THz waves within the 0.1-10 THz frequency range [21]. The mechanism of THz generation in rare gases is still under debate. Based on this, two models have been proposed: the photo-current (PC) model based on tunnel ionization, and the four-wave mixing (FWM) model based on third-order nonlinear effects [22–27]. When a strong electric field is applied to a semiconductor, electrons initially in the VB undergo tunneling transitions to higher bands or nonlinear motion within the bands. Through Bragg reflection, the accelerated electrons, as they reach the Brillouin zone, exhibit BO, gaining energy from the electric field and reaching energy levels within the CB. Enhanced radiation in the THz frequency range [28] occurs when the momentum of the electrons matches the lattice momentum. Notably, THz emission has been observed from superlattices [29] utilizing ultrashort laser pulses, particularly when the laser field strength is not excessively high. In such cases, the laser field is insufficient to induce interband tunneling of electrons, and THz radiation arises from transitions between Wannier-Stark ladder states.
The generation of THz waves in solids is intricately connected to the ultrafast dynamics of electrons engaged in intraband Bloch motion. This phenomenon, known as dynamic localization, is similar to BO and occurs when electrons undergo periodic motion, allowing them to return to their initial state under the influence of an alternating current (AC) electric field. In contrast to BO, which can occur at any field strength of a DC field, dynamic localization only takes place at specific field strengths of an AC field. This phenomenon was initially discovered by Dunlap [30,31] and plays a crucial role in nonlinear electron transport. It occurs when the ratio of $edE_{0}$ to $\hbar \omega$ corresponds to a root of the ordinary Bessel function of order 0: $J_{0}(edE_{0}/\hbar \omega )=0$ [32]. Under this condition, electrons become localized, leading to the suppression of the THz emission signal.
In this paper, we investigate THz radiation in semiconductors by subjecting a one-dimensional typical two-band semiconductor structure to the fundamental laser field and its second harmonic. We calculate the dependence of THz yield on the relative phase of the two-color fields and analyze the THz radiation mechanism within the semiconductor. Subsequently, we solve the SBEs to calculate THz yields under varying driving laser field strengths, establishing the connection between dynamic localization effects and the minima in THz yield. We also examine the influence of dephasing on dynamic localization and THz yield. To assess the impact of polarization effects between bands, we solve the single-band Boltzmann transport equation for comparative analysis. Finally, we propose a two-color method to enhance THz radiation efficiency in semiconductors. This work provides the theoretical basis for joint measurements [33] of THz and HHG [4,7,8] in semiconductor materials, enabling the simultaneous characterization of multi-timescale electronic dynamics within semiconductors.
2. Method
The band structure can be obtained by solving the Schrödinger equation within a single-electron approximation
Here, $m$ represents the mass of the electron, and $\mathbf {L}$ corresponds to the macroscopic large size of the crystal. The periodic potential is denoted by $v(\mathbf {r})$ and it satisfies the condition $v(\mathbf {r}+\mathbf {d})=v(\mathbf {r})$ with $\mathbf {d}$ being the lattice constant. According to the Bloch theorem, the Bloch function can be expressed as follows:
The eigenenergies $\epsilon ^\lambda _\mathbf {k}$ determine the dispersion relation between the electron energy and the crystal momentum $\mathbf {k}$ within the $\lambda ^{\mathrm {th}}$ band. When subjected to an external time-varying laser field E(t), the time evolution of the crystal momentum can be described by:
A set of coupled differential equations that govern the coupled dynamics of electrons and holes under the optical polarization between different bands are known as the semiconductor Bloch equations (SBEs). In general, the SBEs can be formulated by starting from a many-body approach within the two-band approximation. In the regime of extremely nonlinear optics, the Coulomb interaction among the photoexcited carriers is very weak, typically on the order of the exciton binding energy. Therefore, in this work, we can neglect the Coulomb interaction. By solving the following SBEs, both the optical interband polarization $p_{\mathbf {k}}$ and the occupation $n_{\mathbf {k}}^{e(h)}$ of electrons (holes) can be calculated :
The macroscopic current $J$ and the polarization $P$ induced by the motion of electron (hole) within the bands can be expressed as:
and $v_{\mathbf {k}}^{\lambda }=\nabla _{\mathbf {k}}\epsilon _{\mathbf {k}}^{\lambda }$ is the group velocity in the $\lambda ^{\mathrm {th}}$ band, which is defined as the gradient of the energy dispersion $\epsilon _{\mathbf {k}}^{\lambda }$ to the wave vector $\mathbf {k}$. The high harmonic spectrum from the semiconductor structure can be obtained from: where the interband and the intraband emission can be expressed as $I_{\mathrm {pol}}=|\omega ^{2}P(\omega )|^{2}$ and $I_{\mathrm {curr}}=|\omega J(\omega )|^{2}$, respectively.To investigate the motion of electrons within a single energy band while disregarding the influence of interband polarization, we solved the Boltzmann transport equation ignoring the scattering effect:
The value of $n(\mathbf {k},t)$ can be obtained by solving Eq. (10). To determine the macroscopic current, we integrate $n(\mathbf {k},t)$ over the wavevectors across the entire Brillouin zone.
3. Results and discussion
We determine the highest VB and the lowest CB involved in SBEs by solving the single-electron Schrödinger equation with a one-dimensional periodic potential $V(x)=-0.37[1+\cos (2\pi x/d)]$, where the lattice parameter $d=8$ a.u. The typical band structure is exposed to a laser field with a trapezoidal envelope comprising two ascending cycles, six platform cycles, and two descending cycles. The laser pulse has a duration of 106.85 fs, a wavelength of 3200 nm, and a field strength of 0.003 a.u. The corresponding high harmonic spectrum is shown in Fig. 2(a). In the spectrum, the harmonics generated through interband polarization are represented by the solid black line, while the harmonics resulting from intraband macroscopic current radiation are represented by the solid red line. The green dashed line depicts the total harmonic spectrum.
Fig. 2. (a) The HHG spectrum of interband radiation is represented by the solid black line, the solid red line represents intraband radiation, and the dashed green line represents the total harmonic spectrum. (b) The solid black line and the dashed red line represent the HHG spectra under single-color and two-color field driving, respectively. The inset shows the THz spectrum.
The typical two-band structure has a minimum band gap of 3.2 eV, which corresponds to the 10th-order harmonic in the HHG spectrum. We focus on the harmonics below the 0.2nd order, which translates to a 17 THz frequency. Figure 2(a) reveals that THz and HHG below the band gap energy are primarily contributed by the intraband mechanism, which reflects the dynamic characteristics of electrons within the bands. However, HHG above the band gap energy is mainly contributed by interband polarization.
We simplify the crystal as a one-dimensional band structure under the tight-binding approximation, which is symmetric in momentum space. As a result, the symmetric nature of the system leads to the cancellation of intraband macroscopic currents generated by a monochromatic field, resulting in low THz radiation efficiency. To address this issue, a similar approach to atomic systems is employed. The radiation efficiency of THz can be enhanced by introducing asymmetry to the laser field through the addition of a second harmonic with an electric field strength of $3\times 10^{-5}$ a.u. This modification effectively alters the asymmetry of the laser field and enhances the emission efficiency of THz. As depicted in the inset of Fig. 2(b), the THz generated by the two-color field (dashed red line), is three orders of magnitude stronger than the single-color field (solid black line). Additionally, the two-color field induces even-order harmonics. However, due to the relatively weak intensity of the second harmonic, which is only $10^{-4}$ times the electric field intensity of the fundamental frequency field, there is no significant influence on the odd-order harmonics. Breaking the symmetry of semiconductors requires a much lower second harmonic field strength compared to changing the symmetry of atomic systems. This is because the motion of electrons within the band, influenced by dispersion, leads to an accumulation of relative phase in the two-color field, making it easier to break the symmetry.
The generation of THz radiation involves multiple mechanisms, each associated with different optimal phases of a two-color laser field for efficient THz emission. For instance, the FWM model postulates that THz radiation results from higher-order nonlinear effects. In Eq. (11), a two-color laser field interacts with a material, giving rise to nonlinear mixing and the generation of new frequencies, including THz radiation [25,26].
However, the PC model in Eq. (12) suggests that the THz radiation is directly proportional to the residual current, with the optimal phase for the THz radiation being $\pi /2$. Figure 3 illustrates the modulation of THz yield as a function of the relative phase delay between the two-color field. It shows that the THz yield oscillates four times within a period of the fundamental frequency field, indicating that the optimal phase delay for THz radiation is $\pi /2$. According to this model, the macroscopic nonlinear current generated by the motion of electrons within the energy band contributes to THz radiation, as well as HHG below the bandgap.
Fig. 3. THz yield as a function of the phase delay of the two-color field, presented from bottom to top. The peak field strengths of the fundamental field are as follows: 0.003 a.u., 0.004 a.u., 0.005 a.u., 0.006 a.u., 0.007 a.u., 0.008 a.u., 0.009 a.u., 0.01 a.u., and 0.011 a.u. The peak field strength of the second harmonic is 0.0001 a.u.
In Fig. 4, we investigated the modulation of THz yield with the strength of the fundamental frequency field and the amplitude of second harmonic field is 0.0001 a.u. The THz yield initially rises as the fundamental frequency field strength increases. Within the range of fundamental field strengths from 0.001 a.u. to 0.004 a.u., the THz yield shows an approximately quadratic correlation with the fundamental field strength. This aligns with the relationship between THz yield and fundamental field strength in the PC model [22].
Fig. 4. The dependence of THz yield on the strength of the fundamental frequency field calculated with different CEPs of the two-color field, the black dashed lines represent the results with CEP = 0, while the red solid lines indicate CEP = 0.5$\pi$. In (a), (b), and (c), the dephasing times are $T_{2}=10.68$ fs, $T_{2}=5.34$ fs, and $T_{2}=2.67$ fs, respectively.
When an AC field is applied to a semiconductor, electrons in the VB undergo tunnel ionization into the CB, forming electron-hole pairs. As these pairs are accelerated by an electric field towards the Brillouin zone boundary, their behavior depends on the energy bandgap. If the energy bandgap between the higher CB and the lowest CB at the boundary is greater than the energy bandgap between the highest VB and the lowest CB, there is a higher probability of electrons re-entering their original energy band from the opposite side rather than transitioning to a higher energy band. Within this band, they exhibit periodic back-and-forth motion, leading to BO, which result in intraband nonlinear electron motion and THz radiation. According to Eq. (4), the momentum is determined by vector potential. When the laser field strength is below 0.004 a.u., the nonlinear motion remains confined to one side of the Brillouin zone center ($\Gamma$ point). The electron group velocity maintains the same direction, leading to an increase in THz yield from intraband current, as depicted by dashed black lines in the region corresponding to 0.001 a.u. to 0.004 a.u. in Fig. 4.
When the laser field strength exceeds 0.004 a.u., electrons can achieve a momentum that extends beyond the Brillouin zone boundary. They cross the boundary, change the direction of their group velocity, and enter the Brillouin zone from the opposite side, initiating BO. As the group velocity changes direction, according to Eq. (8), the nonlinear current responsible for radiation decreases, leading to a reduction in THz yield. This reduction continues until the frequency of BO resonates with the driving AC field frequency, resulting in dynamic localization and causing a minimum THz yield.
Dynamic localization occurs when the field strength satisfies the zeroth-order Bessel function, $J_{0}(eEd/\omega )=0$ [30,35]. This represents a resonance between Stark oscillations with the Bloch frequency ($eEd$) and AC electric field oscillations at the frequency ($\omega$), resulting in localized electron behavior. This localization effect leads to minimal THz yield. The first two solutions for $J_{0}(eEd/\omega )=0$ are 2.45 and 5.5, corresponding to peak field strengths of 0.00427 a.u. and 0.0097 a.u., respectively. Although the strength of the second harmonic field is weak, it still changes the strength and symmetry of the fundamental frequency field. Therefore, the range of dynamic localization is subject to modulation by the relative carrier-envelope phase (CEP) of the two-color field. In Fig. 4(c), the two modulated THz yield minima are indicated by green arrows. In addition, the degree of dynamic localization has also been modulated by the CEP of a two-color laser field. In Fig. 4, the black dashed lines and the red solid lines correspond to the results calculated when CEP = 0 and CEP = $\pi /2$, respectively. It can be observed that compared to CEP = 0, the results from CEP = $\pi$/2 have relatively smaller THz yields at the first two minima. So, we can further control the range and extent of dynamic localization by changing the symmetry of the two-color field.
In Fig. 4, the THz yield exhibits oscillations as the driving field strength increases, when the peak driving field strength exceeds 0.004 a.u. To investigate the relationship between THz yield oscillations and dephasing effects, we introduced the dephasing time, denoted as $T_{2}$. Interactions between electrons and the lattice, as well as interactions between electrons, can result in shorter dephasing time. A longer dephasing time suggests that carriers spend more time in a coherent state. We varied the dephasing time $T_{2}$ to observe its influence on THz yield oscillations. When the dephasing time equals one laser field period (10.68 fs), the THz yield minima indicated by green arrows disappear. However, when we reduce the dephasing time to $1/4$ of the laser period (2.67 fs), the THz yield minimums generated by dynamic localization become more pronounced.
To further investigate the dynamic localization of electrons, we numerically solve the time-dependent Schrödinger equation, obtaining the trajectory of initially localized electronic wavepackets in coordinate space. The calculation results reveal that when the field strength is equal to 0.003 a.u., 0.008 a.u., and other field strengths that cannot induce dynamic localization, the initially localized electron wave packet undergoes decay, spreading throughout the entire lattice space, as depicted in Fig. 5(a) and (c). However, at field strengths corresponding to the roots of the ordinary Bessel function of order 0, such as 0.00427 a.u. and 0.0097 a.u., electrons become localized, and the region of dynamic localization narrows as the laser field strength increases, as illustrated in Fig. 5(b) and (d).
Fig. 5. Electron trajectories under the laser pulse with field strengths of (a) 0.003 a.u., (b) 0.00427 a.u., (c) 0.008 a.u., and (d) 0.0097 a.u., respectively.
Neglecting polarization and tunneling ionization, we solve the Boltzmann equation in Eq. (10) for localized electron wave packets in the CB. The CEP of two-color laser field is $\pi /2$, and the initial state is a Gaussian wave packet with an initial momentum width of 0.2 a.u. In Fig. 6(a), the solid black line represents the resulting THz radiation. Initially, as the laser field strength increases, the intensity of THz radiation rises. However, it subsequently reaches a minimum value, followed by an increase to the next maximum, and so on. This oscillation in THz yield is a consequence of dynamic localization in the CB under varying strengths of the AC electric field. Dynamic localization occurs at the solutions of $J_{0}(eEd/\hbar \omega )=0$. The dynamic localization calculated for the Boltzmann equation is consistent with Fig. 4.
Fig. 6. The relationship between THz yield and the fundamental frequency field strength in the presence of DC-AC field interactions with band structure. Here, $n$ represents an integer equal to the ratio of $\omega _{B}$ (where $F_{0}$ denotes the strength of DC fields) to the AC field frequency $\omega$.
On the other hand, when interband polarization and transitions are considered in the SBEs, the intensity of interband transitions increases as the driving laser field strength increases. This effect weakens the dynamic localization of electrons within the band. Consequently, when using the SBEs method, it is observed that in regions with a stronger driving laser field, the THz minima generated by dynamic localization become less pronounced, as depicted in Fig. 4.
To enhance THz emission, we add a DC field to an AC field [36]. The THz emission can be enhanced, except the cases where the ratio of the laser field strength and frequency is a root of the Bessel function $J_{n}(edE_{1}/\hbar \omega )$, where $E_{1}$ is the amplitude of AC field, and $n$ is an integer, equals to the ratio of Bloch frequency $\omega _{B}=eE_{0}d (E_{0}$ is the field strength of DC fields) to the AC field frequency $\omega$. We ignore the effects of interband polarization, and tunneling ionization, and focus on solving the Boltzmann equation for localized electron wave packets within CB. When a static electric field $E_{0}$ acts on a periodic potential, it forms a WSL, resulting in the Wannier localized states. If an additional AC field is applied, the Wannier-Stark states exhibit broadening under the n $(n=eE_{0}d /\omega )$ photon resonance. This broadening breaks the Wannier localization. Initially, a localized Gaussian wave packet will propagate with a group velocity $v=(-1)^nd/2J_{n}(eE_{1}d/\omega )\sin (k_{0}d)$ [37,38] determined by the dispersion relation of the highest VB. When the value of $E_{1}$ corresponds to the zero point of $J_{n}(eE_{1}d/\omega )$, the wave packet remains stationary. As we continue to change the value of $E_{1}$, the wave packet periodically undergoes dynamic localization, resulting in the appearance of minima in THz radiation, shown in Fig. 6. When we change the field strength of the DC laser field, $E_{0}$, the relationship between THz yield and the AC field strength $E_{1}$ of $n=1$ is shown. Generally, the THz yield is significantly enhanced, except when dynamic localization occurs, leading to a noticeable suppression of THz emission.
4. Conclusion
In this study, we conducted a theoretical analysis of strong-field emission in semiconductor structures using SBEs. Our calculations demonstrated that the optimal relative phase of THz waves, driven by a fundamental laser pulse and its second harmonic, is $\pi /2$, consistent with the PC model. This finding confirmed that intraband nonlinear current serves as the source of THz radiation. We also observed that the yield of THz radiation exhibits minima at specific laser field strengths, where the ratios between the Bloch frequency and the field frequency correspond to the roots of the Bessel function. This phenomenon is known as dynamic localization. The range and extent of dynamic localization can be further controlled by changing the CEP of the two-color field. When we consider the stronger dephasing effect, electrons can exhibit more dramatic dynamic localization. Additionally, we found that strong interband transitions have a weakening effect on dynamic localization by solving the Boltzmann equation. We further visualized the trajectories of electrons undergoing dynamic localization in coordinate space. Also, a THz-enhanced scheme was proposed to facilitate the measurement in experiments. Our research sheds light on the effects of dynamic localization in the context of modern laser techniques and achieves control of the range and degree of dynamic localization. It provides theoretical support for joint measuring THz and HHG from semiconductors and facilitates simultaneous materials investigations across different time scales.
Funding
National Natural Science Foundation of China (11904341).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. S. Ghimire, A. D. Dichiara, and E. Sistrunk, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011). [CrossRef]
2. M. Korbman, S. Y. Kruchinin, and V. S. Yakovlev, “Quantum beats in the polarization response of a dielectric to intense few-cycle laser pulses,” New J. Phys. 15(1), 013006 (2013). [CrossRef]
3. A. F. Kemper, B. Moritz, J. K. Freericks, et al., “Theoretical description of high-order harmonic generation in solids,” New J. Phys. 15(2), 023003 (2013). [CrossRef]
4. G. Vampa, C. R. McDonald, and G. Orlando, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014). [CrossRef]
5. T. Higuchi, M. I. Stockman, and P. Hommelhoff, “Strong-field perspective on high-harmonic radiation from bulk solids,” Phys. Rev. Lett. 113(21), 213901 (2014). [CrossRef]
6. P. G. Hawkins, M. Y. Ivanov, and V. S. Yakovlev, “Effect of multiple conduction bands on high-harmonic emission from dielectrics,” Phys. Rev. A 91(1), 013405 (2015). [CrossRef]
7. M. Wu, S. Ghimire, and D. A. Reis, “High-harmonic generation from Bloch electrons in solids,” Phys. Rev. A 91(4), 043839 (2015). [CrossRef]
8. L. Liu, J. Zhao, and W. Dong, “Spatial coherence in high-order-harmonic generation from periodic solid structures,” Phys. Rev. A 96(5), 053403 (2017). [CrossRef]
9. Z. Guan, X.-X. Zhou, and X.-B. Bian, “High-order-harmonic generation from periodic potentials driven by few-cycle laser pulses,” Phys. Rev. A 93(3), 033852 (2016). [CrossRef]
10. H. Wang, M. Chini, and S. Chen, “Attosecond time-resolved autoionization of argon,” Phys. Rev. Lett. 105(14), 143002 (2010). [CrossRef]
11. M. Drescher, M. Hentschel, and R. Kienberger, “Time-resolved atomic inner-shell spectroscopy,” Nature 419(6909), 803–807 (2002). [CrossRef]
12. P. M. Paul, E. S. Toma, and P. Breger, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001). [CrossRef]
13. G. Vampa and T. Brabec, “Merge of high harmonic generation from gases and solids and its implications for attosecond science,” J. Phys. B: At. Mol. Opt. Phys. 50(8), 083001 (2017). [CrossRef]
14. S. Ghimire, G. Ndabashimiye, and A. D. Dichiara, “Strong-field and attosecond physics in solids,” J. Phys. B: At. Mol. Opt. Phys. 47(20), 204030 (2014). [CrossRef]
15. F. Bloch, “Über die quantenmechanik der elektronen in kristallgittern,” Z. Phys. 52(7-8), 555–600 (1929). [CrossRef]
16. C. Zener, “A theory of the electrical breakdown of solid dielectrics,” Proc. R. Soc. Lond. A 145(855), 523–529 (1934). [CrossRef]
17. E. E. Mendez and G. Bastard, “Wannier-Stark ladders and Bloch oscillations in superlattices,” Phys. Today 46(6), 34–42 (1993). [CrossRef]
18. H. Wannier, Elements of Solid State Theory (Cambridge University Press, 1959).
19. S. R. Wilkinson, C. F. Bharucha, and K. W. Madison, “Observation of atomic Wannier-Stark ladders in an accelerating optical potential,” Phys. Rev. Lett. 76(24), 4512–4515 (1996). [CrossRef]
20. E. N. Osika, A. Chacón, L. Ortmann, et al., “Wannier-Boch approach to localization in high-harmonics generation in solids,” Phys. Rev. X 7(2), 021017 (2017). [CrossRef]
21. D. M. Mittleman, “Twenty years of terahertz imaging,” Opt. Express 26(8), 9417–9431 (2018). [CrossRef]
22. K. Y. Kim, J. H. Glownia, A. J. Taylor, et al., “Terahertz emission from ultrafast ionizing air in symmetry-broken laser fields,” Opt. Express 15(8), 4577–4584 (2007). [CrossRef]
23. K. Y. Kim, A. J. Taylor, J. H. Glownia, et al., “Coherent control of terahertz supercontinuum generation in ultrafast laser-gas interactions,” Nat. Photonics 2(10), 605–609 (2008). [CrossRef]
24. D. J. Cook and R. M. Hochstrasser, “Intense terahertz pulses by four-wave rectification in air,” Opt. Lett. 25(16), 1210–1212 (2000). [CrossRef]
25. X. Xie, J. Dai, and X.-C. Zhang, “Coherent control of THz wave generation in ambient air,” Phys. Rev. Lett. 96(7), 075005 (2006). [CrossRef]
26. J. Dai, X. Xie, and X.-C. Zhang, “Detection of broadband terahertz waves with a laser-induced plasma in gases,” Phys. Rev. Lett. 97(10), 103903 (2006). [CrossRef]
27. C.-W. Chen, Y.-S. Lin, and J. Y. Huang, “Generation and spectral manipulation of coherent terahertz radiation with two-stage optical rectification,” Opt. Express 16(18), 14294–14303 (2008). [CrossRef]
28. A. A. Ignatov, E. Schomburg, and J. Grenzer, “Thz-field induced nonlinear transport and DC voltage generation in a semiconductor superlattice due to Bloch oscillations,” Z. Phys. B: Condens. Matter 98(2), 187–195 (1995). [CrossRef]
29. C. Waschke, H. G. Roskos, and R. Schwedler, “Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice,” Phys. Rev. Lett. 70(21), 3319–3322 (1993). [CrossRef]
30. D. H. Dunlap and V. M. Kenkre, “Dynamic localization of a charged particle moving under the influence of an electric field,” Phys. Rev. B 34(6), 3625–3633 (1986). [CrossRef]
31. D. H. Dunlap and V. M. Kenkre, “Effect of scattering on the dynamic localization of a particle in a time-dependent electric field,” Phys. Rev. B 37(12), 6622–6631 (1988). [CrossRef]
32. W. Yan, S. Bao, X. Zhao, et al., “Dynamic localization versus photon-assisted transport in semiconductor superlattices driven by DC-AC fields,” Phys. Rev. B 61(11), 7269–7272 (2000). [CrossRef]
33. D. Zhang, Z. Lü, and C. Meng, “Synchronizing terahertz wave generation with attosecond bursts,” Phys. Rev. Lett. 109(24), 243002 (2012). [CrossRef]
34. D. Golde, T. Meier, and S. W. Koch, “High harmonics generated in semiconductor nanostructures by the coupled dynamics of optical inter- and intraband excitations,” Phys. Rev. B 77(7), 075330 (2008). [CrossRef]
35. O. M. Yevtushenko, “Localization and delocalization of electrons in narrow-band semiconductors under the action of strong time-dependent and constant electric fields,” Phys. Rev. B 54(4), 2578–2585 (1996). [CrossRef]
36. A. Zhang, L. Yang, and M. M. Dignam, “Influence of excitonic effects on dynamic localization in semiconductor superlattices in combined DC and AC electric fields,” Phys. Rev. B 67(20), 205318 (2003). [CrossRef]
37. M. Holthaus, “Collapse of minibands in far-infrared irradiated superlattices,” Phys. Rev. Lett. 69(2), 351–354 (1992). [CrossRef]
38. M. Holthaus and D. W. Hone, “Localization effects in ac-driven tight-binding lattices,” Philos. Mag. B. 74(2), 105–137 (1996). [CrossRef]
