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Abstract
High order harmonic generation (HHG) in semiconductors opens a new frontier in strong field physics and attosecond science. However, the underlying physical mechanisms are not yet fully understood and lively debated. Here, we identify and discuss carrier-wave population transfer as a novel and important dynamical effect. We find that the interband excitation occurs in an extremely short time window due to the intraband motion. Our analysis based on this finding allows for a physically intuitive interpretation of the anomalous carrier-envelope phase dependence observed in HHG from MgO and to understand the dominant role of the interband polarization as reported in a series of recent semiconductor HHG experiments. Motivated by the discovered coupling mechanism, we demonstrate that the interband excitation can be controlled by an appropriately tailored two-color field. An ultrabroad supercontinuum spectrum covering the entire plateau region can be generated which directly creates an isolated-attosecond pulse even without phase compensation. Our results provide remarkable insight into the basic physics governing the sub-cycle electron motion with significant implications for the generation of isolated-attosecond light pulses in semiconductor materials.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
More than a decade ago, it was pointed out that whenever the energy associated with the light intensity, e.g., the Rabi energy and the Bloch energy, becomes comparable to or even larger than the photon energy of the laser field, the laws of “traditional” nonlinear optics fail and a new regime is entered [1]. This regime has been referred to as “extreme nonlinear optics” or “carrier-wave nonlinear optics”. Carrier-wave Rabi flopping (CWRF) effect has been demonstrated in semiconductor GaAs [2] and potassium atoms [3] for near resonant interactions.
Recently, high-harmonic generation (HHG) arising from strong non-resonant excitation of semiconductors has attracted great attention due to the significant opportunities for the applications in attosecond pulse generation and all-optical reconstruction of electron band structure of semiconductors [4–15]. Moreover, the idea of sub-cycle manipulation has been extended from atoms and molecules to solids, allowing to monitor and steer charge carriers on the attosecond time scale [16–26]. Approaching and exploiting these frontiers further requires an improved insight into the underlying physics of the carrier dynamics in semiconductors.
The physical mechanism for HHG in semiconductors is significantly more sophisticated than that in atomic systems due to their complexity and diversity. Besides the interband polarization, the intraband current also contributes to HHG in semiconductors. To date, which mechanism dominates is still highly debated [4–15, 20–27]. A series of theoretical and experimental investigations have demonstrated that the interband contribution dominates the harmonics above band gap for ZnO, MgO, and GaAs exposed to mid-IR fields [6–8, 23, 27–29]. However, a contradictory fact is that the Bloch frequency is usually much larger than the Rabi frequency in most semiconductors driven by the same laser field, which means that the electron traverses a large fraction of the Brillouin zone on a short time scale compared to that of the interband transition [4]. As a result, the role of intraband mechanism should be nontrivial in semiconductors. This conclusion has been underpinned by recent experimental findings. It has been demonstrated in [25] that though intraband mechanism itself accelerates carriers only within the same band, the coupling between the two mechanisms can significantly influence the carrier injection from the valence into the conduction band, and the light-matter interaction in semiconductor GaAs is actually dominated by the intraband motion even for resonant excitation. Since the carrier excitation is critical for almost all strong-field phenomena in semiconductors, including the HHG for non-resonant excitation, it is imperative to clarify the exact role of intraband motion on HHG, especially for those cases where the dominance of the interband polarization has been demonstrated [6–8,23,27–29].
In this work, we show that a novel carrier-wave population transfer (CWPT) occurs for the intense non-resonant excitation of semiconductors in the extreme nonlinear optical regime. The carrier excitation from valence band to the conduction band is confined within an extremely short time window due to the intraband motion. We identify that it is the intraband motion that makes the interband polarization become the dominant contribution to the HHG in many semiconductors [6–8,23,27–29] and explains the experimentally observed anomalous carrier-envelope phase (CEP) dependence when driven by few-cycle ultrashort pulse [23]. Most importantly, we show that the discovered physical mechanism can be applied to control the carrier excitation. By applying a two-color field, an ultrabroad supercontinuum spectrum covering the entire plateau region can be generated which creates an intense isolated attosecond pulse even without phase compensation.
2. Comparison with experiments and discussion
To explore how intraband motion influences the interband dynamics on sub-optical-cycle attosecond time scale, we take the recent HHG experiment in single-crystal MgO driven by a waveform-controlled two-cycle laser pulse as an example, where the measured harmonic spectrum has been demonstrated to be dominated by interband polarizaton and shows an anomalous CEP dependence [23]. The electric field has the form . The field amplitude is E0 = 1.3 V/Å, the laser frequency ω0 = 1.109 fs−1 corresponds to a central wavelength wavelength λ = 1.7 μm, τp = 2T0 is the full width at half maximum (FWHM) of the pulse and T0 is the laser period T0 = 2π/ω0, and ϕCEP is the CEP.
The semiconductor Bloch equations are employed to investigate the coupled inter- and intraband dynamics [27]. An advantage of this method is that the intraband motion can be selectively included or neglected in the numerical simulations, which allows one to transparently analyze its role on the HHG from semiconductors. Moreover, the total emission Irad, the emission due to the interband polarization , and that due to the intraband current can be clearly distinguished. The band structure parameters are taken from Ref. [23]. As shown in Fig. 1(a), the slope of the photon energy versus the CEP with a π periodicity can be clearly seen, which reproduces well the experimental result shown in Fig. 2(b) of Ref. [23] and thus justifies the validity of our simulations. We compare the two- and three-band simulations [see Figs. 1(a) and 1(b)], and the results are nearly the same, demonstrating that the CEP dependent signals observed in Ref. [23] are quite universal, and is not related to the excitation to higher conduction band as in Ref. [29,30].
Fig. 1 Simulated CEP-dependent HHG spectra for MgO with field strength 1.3 V/Å and excitation carrier wave central wavelength 1.7 μm, corresponding to a laser period T0 = 5.665 fs. We use a temporal Gaussian envelope with a FWHM τp = 2T0 and a dephasing time of T2 = T0/4. (a) with two-band model, (b) with three-band model, and (c) considering only interband transitions.
Fig. 2 (a), (c), (e) The total HHG spectrum Irad (black dash line), interband polarization (blue solid line), and intraband current (red solid line) for coupled inter- and intraband dynamics, and pure interband HHG (green solid line) when the intraband motion is artificially switched off. (b), (d), (f) The corresponding time-dependent population nC1[(K(t), t] for coupled (blue line) and pure interband (green line) motion. The red dash and green dot-dash lines in (b), (d), and (e) show the normalized electric field and vector potential, respectively. with ϕCEP = 0 and a field strength 1.3 V/Å. (a) and (b) show the results for τp = 2T0 and field strength 1.3 V/Å, (c) and (d) are same with (a) and (b) but for τp = 10T0. (e) and (f) are same with (c) and (d) but for ΩR = 0.7ω0 and ΩB = 8.45ω0. The dotted grey vertical lines in (a), (c), and (e) represent the minimum band gap at the center of the Brillouin zone. The blue dots on the curves in (b), (d) and (f) denote moments when the electrons pass through the Γ point. The magenta dots in (f) denote moments when the electrons after Bragg-reflection at the boundary of the first Brillouin zone pass through the Γ point again.
Figure 2(a) shows HHG spectra for CEP ϕCEP = 0. The spectrum is very complex and does not show a clear odd harmonic structure in the plateau region. The efficiency of the interband HHG (blue solid line) in the plateau region is higher by at least 2 orders of magnitude than that of the intraband HHG (red solid line). The total HHG emission (black dash line) in the plateau is dominated by the interband polarization, which is consistent with the conclusion of Ref. [23]. Surprisingly, when the intraband motion is artificially turned off, the efficiency of the pure interband HHG decreases dramatically by about 5 orders of magnitude, and only the 1st and 3rd harmonics can be generated with the same laser parameters (see green solid line in Fig. 2(a)). Correspondingly, the CEP-dependent energy shift hardly occurs, see Fig. 1(c). In fact, such a significant enhancement of the interband HHG by the intraband motion is not limited to MgO but also exists in other semiconductors such as GaAs, ZnO, etc. [6–8,27]. However, to the best of our knowledge, the underlying physics of why and how the intraband motion influences the efficiency of the interband HHG has not been explored so far.
To reveal the underlying physics, we show in Fig. 2 (b) the time-dependent population of the first, i.e., the energetically lowest, conduction band (CB) nC[K(t), t] [22]. According to the Bloch acceleration theorem [31], the crystal momentum in a given band will change as
where k = K(t0) is the initial momentum and A(t) is the vector potential. Since most of the electrons are excited near the Γ point, we choose k = 0 as an example here [see blue line in Fig. 2(b)]. The red and green dot-dash lines indicate the normalized electric field and the vector potential, respectively. For comparison, we also present the time-dependent population nC[K(t), t] for pure interband dynamics [see green solid line in Fig. 2(b)]. The population is smoothly increasing with the time-dependent electric field when the intraband motion is neglected. In contrast, when the coupled inter- and intraband dynamics is considered [see blue solid line in Fig. 2(b)], a novel dynamics called CWPT is formed. The population is nearly constant near E(t) ≈ 0, whereas undergoes very rapid increase within a very short time window (≈ 260 as) when the vector potential is approximately equal to 0. At the moments of rapid excitation, the electrons pass through the Γ point and the electric field reaches the extrema. Similar phenomenon has also been found very recently for resonant exciation cases [22,25]. This effect can be attributed to the fact that at the moment E(t) ≈ 0, the vector potential of the electric field reaches the extreme values. According to the Eq. (1), the crystal momentum of the electron will move far away from the Γ point which greatly suppresses interband excitation due to the increase of the band gap. As a result, both the result shown in our work for non-resonant excitation in MgO and that in the recent experiment for resonant excitation in GaAs [25] demonstrate one fact that intraband motion not only accelerates carriers within the same band, but also significantly confines the excitation from the valence to conduction bands to an attosecond time window at each half-cycle.Figure 2(b) shows that due to the the temporal confinement, the final population in the CB for coupled inter- and intra-band dynamics is significantly lower than that for the pure interband excitation case. This is opposite to the prediction for resonant excitation of Ref. [25]. The reason is that for resonant excitation, Rabi flopping is able to depopulate the CB when neglecting the intraband motion, which reduces the overall density of excited carriers to much lower values than that attainable in the case when both excitation mechanisms are included [25]. For the far-from resonant excitation considered here, Rabi flopping cannot occur, neither with nor without including the intraband motion. In this case, the consequence of the temporal confinement by the intraband motion is reducing the overall photoexcitation. However, counterintuitively, the HHG from the interband polarization is at least five orders of magnitudes stronger with (blue line in Fig. 2(a)) than without (green line in Fig. 2(a)) intra-band motion. This is a surprising finding since it is expected that higher CB population would lead to a higher emission efficiency of HHG. In fact, recent experiment has indeed shown that a higher carrier density does not facilitate the HHG but only reduce it [28]. Our result indicates that the confinement of the interband excitation within an extremely short time window is important for the interband HHG, which benefits the constructive interference among the generated carriers, and enhances the nonlinear interband polarization. So clearly it is the nontrivial coupling of interband and intraband dynamics that leads to a significant enhancement of interband polarization and makes it the dominant contribution to the total HHG.
Another consequence of the temporal confinement of the interband excitation is a broadening of the individual harmonics. To remove the broadening effect induced by the short pulse duration of the driving pulse, we further perform simulations with a long pulse (τp = 10T0). As shown in Fig. 2(d), the temporal excitation confinement by the intraband motion can still be observed, which demonstrates that the CWPT does not significantly depend on the pulse duration, see Figs. 2(b) and (d). A prerequisite for the CWPT is that the Bloch frquency ΩB should be larger than the laser frequency ω0, which means that the CWPT is clearly an extreme nonlinear optical effect. Here the peak Bloch frequency is ΩB ≈ 3.75ω0 and ΩR ≈ 0.31ω0. Well discrete odd-harmonics can be observed (see Fig. 2(c)), and the individual harmonics in the plateau region are much wider than the low-order interband harmonics below the band gap which are induced by multiphoton excitation.
Figure 3(a) presents HHG spectra as a function of the CEP for τp = 10T0. In this case, the CEP-dependent energy shift observed in previous experiments with few-cycle pulses hardly occurs. The reason is that, in spite of the broadening of the individual harmonics by the temporal excitation confinement, there is little overlap and thus interference between adjacent harmonics. When the pulse duration is decreased to τp = 4T0, the harmonics are further broadened due to the short pulse. The tails of adjacent odd-order harmonics would overlap near the positions of even-order harmonics. The resulting interference induces some small peaks between the adjacent odd-order harmonics whose energy depends sensitively on the CEPs of the driving pulses, see Fig. 3(b). The mechanism of this CEP dependence is quite similar to that of the frequency doubling induced by CWRF for resonant excitation, where the high-frequency peak of the fundamental Mollow triplet and the low-frequency peak or the third-harmonic carrier-wave Mollow triplet meet at the position of the second harmonic in an inversion symmetric medium [1,32]. A common feature of both CWPT and CWRF is that the period of such interference-induced CEP dependence is π, rather than 2π.
For a shorter pulse τp = 2T0, the broadening of the individual harmonics is even more prominent such that the tail of the nth-order harmonic can reach the position of the (n ± 2)th harmonic. As a result, obviously, it is the harmonic broadening induced by CWPT in combination with the short pulse duration that induced the interference between different harmonics, which leads to the complex structure of the HHG spectrum shown in Fig. 2(a) and the unique CEP dependence observed in experiment [23] and our simulations, see Fig. 1(a).
When the laser intensity is increased so that both the Rabi frequency and the Bloch frequency are close to or even larger than the laser frequency, multiple-transitions can be observed within each half-cycle, see Fig. 2(f) where ΩR = 0.7ω0, the magenta dots in Fig. 2(f) denote moments when the electrons are reflected back from the boundary of the Brillouin zone and pass through the Γ point again. As shown in Fig. 2(e), a clear harmonic splitting is obtained for ω > 9ω0, which is directly analogous to CWRF observed in resonant excitation of GaAs [2] and of potassium atoms [3] around the third harmonic. Neglecting the intraband motion, CWRF can be observed only for ΩR ≫ ω0 under off-resonant excitation [33]. Here, the CWRF-like splitting of harmonics covers nearly the entire plateau region of the HHG spectrum even for ΩR < ω0, which further confirms the significant enhancement of extreme nonlinear optical effects by the novel CWPT.
3. Isolated attosecond pulse generation
Motivated by the discovered physical mechanism, we further demonstrate that the ultrafast electron dynamics in semiconductors can be controlled by a two-color field. The two-color field is expressed as
Here Ei and ωi (i=1,2) are the peak electric fields and frequencies, respectively. f(t) is the Gaussian envelope and φ is the relative phase between the two laser pulses. The parameters of the strong fundamental field (i=1) are the same as those used in Fig. 2(a), whereas the parameters of the weak controlling pulse are E2 = 0.25E1 and ω2 = (1/3)ω1. As displayed in Fig. 4, both ultrabroad supercontinuum generation and well discrete spectrum can be obtained by varying the relative phase φ, see Fig. 4(a) and Fig. 4(d).Fig. 4 (a) and (b) HHG spectra and the corresponding time-dependent population nC[(K(t), t] for the two-color field with relative phase φ = 0; (c) the temporal profiles of the attosecond pulse generated from supercontinuum spectrum ranging from the 9th to the 21th harmonics without the phase compensation. (d) and (e) same with (a) and (b), but for φ = π
It is clear from the above analysis that the interband excitation occurs mainly when the vector potential A(t) ≈ 0, i.e., when the carriers are located near the Γ point, see Fig. 4(b) and 4(e). However, in contrast to single pulses, A(t) = 0 does not always correspond to the extremum of the electric field of the two-color field. For φ = 0, only at the central peak, A(t) ≈ 0 (see green dash-dot line), the electric field reaches its maximum at the same time and the interband excitation is very strong. For the other half-cycles, either the electric reaches its extremum but the vector potential is nonzero, or the vector potential is zero but the electric field is weak. In both of these two cases, the interband excitation is suppressed. Therefore, the interband excitation is mainly confined to the central half-cycle, which results in the smooth ultrabroad supercontinuum covering almost the entirely plateau region. By filtering the 9th–21th harmonics, an isolated attosecond pulse with a duration of 566.5 as can be generated even without any phase compensation, see Fig. 4(c).
To further understand the HHG and the corresponding control mechanism in semiconductors, we show in Fig. 5 the excitation and emission moments relative to the electric field (red dash line) and the vector potential (green dot-dash line) for a single driving pulse (a) (d), a two-color pulse with relative phase φ = 0 (b) (e), and φ = π (c) (f), respectively. For comparison, we normalize the time-dependent population of conduction band nC(K(t), t) (blue solid lines in Fig. 5(a)–(c)) which indicates the excitation moments. The black lines in Figs. 5(a)–5(c) indicate the emission instants of attosecond pulse generation by simply performing an inverse Fourier transformation of the corresponding spectrum of the 9th–21th harmonics. The time-frequency analyses of the HHG spectra are shown in Fig. 5(d)–5(f), respectively. For the case of a single few-cycle pulse (Figs. 5(a) and 5(d), the parameters are same with those in Fig. 4(a)), a train of three attosecond pulse bursts is obtained. The central burst is the strongest, which originates from the strongest excitation by the central peak of the electric field. This demonstrates that the photons are emitted within about 1/4 optical cycle of the electric field directly folllowing the excitation of electrons into the conduction band (indicated by the black line with arrow). This is quite different from the HHG and attosecond pulse emission in atoms where about 3/4 of an optical cycle is needed that the electron are pulled back and recombine with the core [34]. From the time-frequency analysis, one can see that the HHG and the attosecond pulse mainly originate from the contributions of short trajectories, which is consistent with the case observed in HHG from ZnO [8]. When futher decrese the pulse duration, the satellite attosecond bursts will be suppressed, which means that the amplitude gate adopted in atoms [34,35] also applies to solid HHG.
Fig. 5 Excitation and emission instant analysis for a single driving pulse (a) (d), a two-color pulse with relative phase φ = 0 (b) (e), and a two-color pulse with φ = π (c) (f). (a)–(c) show the electric field (red dash line), the vector potential (green dot-dash line), the time-dependent population of conduction band nC[(K(t), t] (blue solid line), and attosecond pulse generation (black line), respectively. (d)–(e) show the corresponding time-frequency analysis.
For the two-color field with relative phase φ = 0, the excitation at the central peak is enhanced whereas the adjacent submaximal peaks are suppressed, so the excitation and the subsequent photon emission are mainly confined to a single half-cycle of the electric field (see Figs. 5(b) and 5(e)), which results in the smoothed ultrabroad supercontinuum covering the entire plateau region [see Fig. 4(a)] and the isolated attosecond pulse generation [see Fig. 4(c)]. For the case of φ = π, however, two attosecond pulse bursts are generated. As a result, the HHG reveals a feature of periodic modulation. In HHG from atoms, the highest energy emission comes from ionization events driven by the secondary maximum half-cycle electric field. As a result, the supercontinuum is usually located at the cutoff range when driven by a single few-cycle pulse or at the second plateau when driven by a two-color field [36,37]. In contrast, here the supercontinuum spectrum comes from the excitation by the maximum peak of the electric field, which results in a supercontinuum spectrum covering the entire pleateau of the spectrum (see Fig. 4(a)). For φ = π, interband excitation from the central half-cycle is suppressed whereas that from adjacent half-cycles is enhanced. The interference among the photons emitted from different half-cycles induces the discrete spectrum in the plateau region, see Fig. 4(d) and Figs. 5(c) and 5(f), which demonstrates the controllability of the ultrafast carrier dynamics in semiconductors and confirms the validity of our analysis.
4. Conclusions
In conclusion, we report a novel CWPT effect in semiconductors arising from the coupling of the intra- and the interband dynamics. Our study indicates the crucial role of the intraband motion for HHG from semiconductors even when the interband polarization has been proven to dominate the emission. It is identified that the temporal confinement of the interband excitation due to the intraband motion can greatly enhance extreme nonlinear optical effects. Moreover, motivated by the discovered physical mechanism, we demonstrate that the excitation dynamics in semiconductors can be controlled by a two-color field. An ultrabroad smooth supercontinuum covering the entire plateau region can be generated, which directly creates an intense isolated attosecond pulse even without phase compensation. Furthermore, we demonstrate that the generation and control mechanisms of attosecond pulse generation in semiconductors are quite different with those in atoms: (i) in atomic HHG, the ionization process cannot be influenced by electron motion in the continuum state, whereas in semiconductions, intraband motion can dramatically influence the interband excitation. (ii) The attosecond pulse bursts are emitted about 3/4 of an optical cycle after the ionization in atoms, whereas in semiconductor, it is emitted just 1/4 of an optical cycle directly after the electronic excitation. (iii) Due to the excitation confinement by the intraband motion, the interband excitation can be controlled by the relationship between the electric field and the vector potential of the exciting pulse for semiconductors. Our results pave the way toward a more complete understanding of the underlying coupled inter- and intraband dynamics within a sub-cycle attosecond time scale from which applications including the ultrafast control of the electron dynamics and the generation of isolated attosecond sources with high efficiency from solids may greatly benefit.
Funding
National Natural Science Foundation of China (NSFC) (11674209, 11774215, 11674268, 91850209); Open Fund of the State Key Laboratory of High Field Laser Physics (SIOM); “YangFan” Talent Project of Guangdong Province; Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) (231447078, TRR 142 project A07).
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