1. INTRODUCTION

High harmonic generation (HHG) in solids was first examined theoretically [1–5] and has since been demonstrated in a wide range of materials [6–19]. This has laid the foundation for the rapid advancement of attosecond science in condensed matter [20–28], for which HHG is a fundamental process. HHG in solids has also caught attention as a source for ultrashort XUV radiation [10,28] and as a tool to measure ultrafast dynamics and structural properties, such as band structure [12] and the Berry curvature [16,17]. HHG in solids is driven by interband and intraband currents [29,30]. While the interband current is more dominant in wideband materials such as semiconductors [12], HHG in narrow-band dielectrics is driven more by the intraband current [10]. This work focuses on interband HHG in wideband materials.

Although some experimental features can be reasonably well reproduced by numerical models [31–39], a thorough understanding of all the components shaping harmonic spectra is still missing. This inhibits progress in optimizing HHG as a radiation source and in further developing HHG as a diagnostic tool.

The principal mechanism of interband HHG has been clarified by generalizing the saddle-point approach for atomic HHG [40,41] to integrate the interband current derived in the Bloch basis [30,42–45]. Electron and hole are born at the same lattice site in real space by tunnel ionization and quiver in the laser field. When they recollide at some lattice site, a harmonic photon is emitted. Its energy is equal to the bandgap at the crystal momentum of the electron–hole pair at recollision. Despite its merits, the Bloch quasi-classical model falls short of accounting for the lattice structure; quantum mechanics allows recombination of electrons and holes at different lattice sites, as was clearly demonstrated in recent work [15,43,46].

Here we develop a generalized quasi-classical approach that accounts for the lattice structure; this is achieved by transforming the interband current from Bloch to Wannier basis followed by saddle-point integration. The basis change has a substantial effect. The resulting Wannier quasi-classical (WQC) model is found to be in quantitative agreement with quantum calculations. So far, quasi-classical $k$-space analysis has been used to qualitatively investigate strong field effects in gases and in the condensed matter phase; quantitative agreement has not yet been demonstrated. Whether quantitative agreement can be obtained in the Bloch basis remains to be seen; however, the richer physics revealed by the WQC picture indicates that this might not be the case. The more refined WQC picture arises from the fact that the transition dipole moment enters the classical action in the exponent, and therewith, the saddle-point equations. Previously, the Wannier basis has been used in the numerical analysis of HHG [46,47]; a maximally localized Wannier basis [48] also has the advantage of providing a smooth $k$-space gauge for calculating the transition dipole matrix elements [47]. This is of particular importance for treating noncentrosymmetric materials that exhibit complex transition dipoles [36,44,49].

In this work, WQC and quantum calculations are compared for a 1D delta function two-band model solid in the single particle approximation (noninteracting electron–hole pair); parameters are chosen such that the dipole moment and bandgap are representative for semiconductors. The quantitative agreement with quantum calculations suggests that the WQC approach completes the single-body picture for HHG in semiconductors. An electron and hole can ionize and recombine at different lattice sites with a probability determined by the tunneling exponent and Wannier dipole moments; birth and recombination sites are connected by classical trajectories; quantum effects are included by a quadratic expansion of the classical action about the classical trajectories. Beyond that, WQC analysis allows unprecedented insight into the real-space aspects of tunnel ionization in solids; it gives access to the tunnel ionized wave function in real space and therewith, to the birth location of the electron–hole pair. While Bloch quasi-classical analysis describes an electron–hole pair by a single trajectory, WQC analysis describes it by a swarm of weighted trajectories. The increased accuracy comes at the cost of higher computational load.

With a complete understanding of the single electron picture of HHG in two-band semiconductors, it is possible to begin looking at more complex dynamics. On a fundamental level, the WQC approach presents a novel platform from which alternative pathways can be developed for modeling noise and few electron–hole dynamics in solids; classical stochastic equations are easier to handle than quantum systems coupled to stochastic heat baths; as propagation from initial to final Wannier wave packet is done by classical trajectories, the space in between does not need to be resolved, in contrast to a full quantum approach. Thus, WQC analysis holds the promise to provide better scalability and a more intuitive understanding of quantum dynamics in semiconductors. Finally, multiband dynamics can be introduced into WQC analysis by adapting the approach developed in Ref. [18], which also showed that higher conduction bands mostly dominate higher plateaus, and to a lesser extent influence the first plateau of the harmonic spectrum.

One of the ultimate goals of HHG spectroscopy of solids is to extract structural and dynamic data from harmonic spectra. This will be very difficult without simple models. WQC analysis establishes the capacity to capture quantum dynamics in terms of classical trajectories in a reasonably quantitative fashion; this makes it suitable as a diagnostic tool for HHG spectra obtained from experiments and numerical quantum analysis.

More generally, our analysis opens an avenue for modeling quantum dynamics of wave packets by propagating classical trajectories. This is potentially relevant for a wide spectrum of applications, ranging from strong-field physics to transport phenomena [50,51] and coherent control [52,53].

2. THEORY

A. Two-Band WQC Model

Our formalism is developed for a 3D, two-band model. We first summarize the derivation of HHG in the Bloch basis [30]; it starts from the time-dependent Hamiltonian $H(t) = {H_0} + {\bf x} \cdot {\bf F}(t)$; ${\bf F}(t)$ represents the laser field; ${H_0}$ is the unperturbed lattice Hamiltonian with Bloch eigenstates ${\Phi _{m,{\bf k}}}({\bf x}) = 1/\sqrt V {u_{m,{\bf k}}}({\bf x})\exp (i{\bf k} \cdot {\bf x})$ and with energies ${E_m}({\bf k})$ in band $m$ with crystal momentum ${\bf k}$; the band index $m = v,c$ refers to the valence and the conduction band, respectively; ${u_{m,{\bf k}}}$ is the periodic part of the Bloch function, $\langle {\Phi _{m,{\bf k}}}|{\Phi _{m,{\bf k}}}\rangle = 1$, and $\langle {u_{m,{\bf k}}}|{u_{m,{\bf k}}}\rangle = \upsilon$. Finally, $V = N\upsilon$ is the volume of the solid, with $N$ and $\upsilon$ the number and volume of primitive unit cells. Hartree atomic units are used, unless otherwise noted.

In the presence of the laser field the wave function becomes time-dependent. In the length gauge it is represented as

In the following, we will translate HHG, as described by the interband current of Eq. (2), from $k$ space to real space by using Wannier functions. The Bloch and Wannier basis functions are connected by a Fourier transform according to [55]

In order to translate the interband current [Eq. (2)] into real space, the Bloch functions in the transition dipole moment [Eq. (3)] are replaced by the Wannier functions with the help of relation (4a). This leads to

Inserting Eq. (6) into Eq. (2), the interband current follows as

B. Saddle-Point Integration

The integrals in Eq. (7b) are solved by saddle-point integration. The saddle-point equations,

The set of Eq. (8) is solved for a linearly polarized laser field ${\bf F} = F\hat{\boldsymbol x}$; further, ${\bf A} = A\hat{\boldsymbol x}$ and ${k_x} = k$. The solutions of the saddle-point equations are denoted by $t^\prime = {t_b} + i\delta$, $t = {t_r}$, ${{\bf k}_s}$. For $\delta \ll 1$, Eq. (8a) can be solved analytically; it determines the saddle-point momentum ${{\bf k}_s} = ({k_s},{k_{\textit{ys}}},{k_{\textit{zs}}}) = (A({t_b}) - A({t_r}),0,0)$, as well as

The two remaining saddle-point equations, (8b) and (8c), determine ${t_b}$ and ${t_r}$. They have to be solved numerically for each possible birth site ${{\bf x}_l}$ and recombination site ${{\bf x}_j}$; for instance, by running through ${t_b}$ and finding all ${t_r}({t_b})s$ that fulfill Eq. (8c). From those, the pairs $[{t_b},{t_r}](\omega)$ are selected that produce a given harmonic $\omega$ via Eq. (8b). The physical implications of the saddle-point equations are discussed at the end of this subsection.

Next, the integrand of Eq. (7b) is evaluated at the saddle point, where the small imaginary birth time determines the tunneling exponent. Further, the phase $\varphi$ is expanded to second order, which gives the multivariate Gaussian integral,

The propagator [Eq. (12)], together with the saddle point [Eq. (8)], and the interband current [Eq. (7a)] represent the WQC description of HHG in semiconductors. They reveal a complete and detailed picture of the physical mechanisms driving HHG in real and reciprocal space, summarized in Figs. 1(a) and 1(b), respectively. The empty circles in Fig. 1(a) represent the centers of the atomic unit cells ${{\bf x}_l}$, where $l = ({l_x},{l_y})$ in the 2D schematic. A Wannier basis function is located at each center. Initially, all Wannier sites of the valence band are filled. As all lattice sites are identical, it is sufficient to investigate ${{\bf x}_l} = 0$; see Eq. (5) above. Following the notation of our calculation, we chose indices $l,j$ to represent birth and recombination sites, respectively. HHG proceeds in three steps:

 

Fig. 1. Schematic of the WQC picture of interband HHG. (a) Real space picture for a model 2D lattice; empty circles denote centers of atomic unit cells at which Wannier basis functions are located. Distances shown refer to relative distance between each electron–hole pair. Classical and QM processes are indicated. The dotted arrows point to the probability amplitudes of the individual processes. In addition, the phase ${\chi _1}$ picked up along the classical trajectory is indicated. HHG takes place in three steps. (1) An electron initially at the valence Wannier site ${{\bf x}_0}$ is born at ${{\bf x}_0} + {{\bf x}_l}$, creating an electron–hole pair $l$ lattice sites apart (red arrow), with ionization amplitude $\propto {\bf d}_l^ * \exp [- {t_x}]$. (2) Then it propagates in the laser field along the classical trajectory ${\boldsymbol \xi}({t_b},{t_r})$; the green shaded area indicates the quasi-classical contribution ${\bf g}({t_b},{t_r})$ that comes from the Gaussian expansion of the propagator about the classical trajectory ${\boldsymbol \xi}$. (3) Electron and hole revisit each other and recombine $j$ lattice sites apart with ${{\bf d}_j}$ the recombination dipole (blue arrow). (b) ${\bf k}$-space picture, full and empty circles in valence band indicate filled states and empty states (holes), respectively; (1) electron–hole pairs are born at the $\Gamma$ point (${\bf k} = 0$); (2) the laser field drives them in reciprocal space (green arrow); (3) they recombine at some different ${{\bf k}_s}$.

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3. RESULTS

For the remainder of the paper, the WQC approach and its physical significance are explored within a 1D model system. In this case the interband current, WQC propagator, and probability amplitude reduce to scalars, namely, ${\tilde j_{\textit{er}}}$, ${T_{\textit{jl}}}$, and ${P_{\textit{jl}}}$. Specifically, we use a 1D delta function model potential, $V(x) = \Omega \sum\nolimits_{n = - \infty}^\infty \delta [x - (n + 1/2)a]$ with unit cell size $a$ and barrier penetration parameter $\Omega$. This model solid represents inversion symmetric semiconductors, for which the intraband dipole moment (Berry connection) is zero in a maximally localized Wannier basis. Details of the delta function model are given in Appendix B. For the investigated parameters, the bandgap is well approximated by the nearest neighbor dispersion $\varepsilon (k) = {E_g} + \Delta [1 - \cos (ka)]$, where ${E_g}$ is the minimum bandgap and $2\Delta$ represents the bandwidth. We chose $a = 7$ and considered two values $\Omega = 0.5,1.5$ to model a weakly and tightly bound semiconductor, respectively. The corresponding bandgap parameters are ${E_g} = 0.141,0.269$ and $\Delta = 0.269,0.17$. Finally, for all runs we use a dephasing time ${T_2} = {T_0}/2$ so that only returns within a single cycle are relevant.

In Fig. 2 the exact (QM) harmonic spectrum, as obtained from numerical integration of Eq. (2), is compared with the Wannier quasi-classical solution: Eqs. (7a), (8), (9), and (12a). For the exact approach, we use $F(t) = {F_0}\sin ({\omega _0}t)\exp (- {(t/\tau)^2})$, where ${F_0}$ is the maximum field strength, and the pulse duration, $\tau = 40{T_0}$, is long enough to approach the continuous wave (cw) limit; ${\omega _0}$ is the laser center frequency, and ${T_0} = 2\pi /{\omega _0}$ denotes the optical cycle. For the highest field strength considered, an electron born at the $\Gamma$ point explores only ${\sim}60\%$ of the BZ and does not reach the next bandgap at the edge of the BZ to transition to the second conduction band; this justifies the approximation to include only two bands. We plot the harmonic intensity $|{h_n}{|^2} = \int_{{\omega _ -}}^{{\omega _ +}} d\omega |{\tilde j_{\textit{er}}}(\omega {)|^2}$ integrated over the frequency interval ${\omega _ \pm} = (n \pm 1/2){\omega _0}$.

 

Fig. 2. Harmonic yield $|{h_n}{|^2}$ versus harmonic order $n$; $a = 7$, ${T_2} = {T_0}/2$_. (a)–(b) Empty blue circles with lines (exact) and filled blue circles (WQC) refer to $\Omega = 0.5$, ${\omega _0} = 0.01425$ ($\lambda = 3.2\,\,\unicode{x00B5}\text{m}$), and ${F_0} = 0.0025$ in (a) and ${F_0} = 0.0015$ in (b); empty red squares connected by lines (exact) and filled red squares (WQC) refer to $\Omega = 1.5$, ${\omega _0} = 0.0285$ ($\lambda = 1.6\,\,\unicode{x00B5}\text{m}$), and ${F_0} = 0.008$ in (a) and ${F_0} = 0.005$ in (b); lines are used to guide the eye.

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For the WQC calculation, we assume the cw limit, $F(t) = {F_0}\sin ({\omega _0}t)$, in order to facilitate interpretation of the results. Equation (12a) has been derived for finite pulses employing the Fourier transform. For a transition to the cw limit, the Fourier transform has to be replaced by a Fourier series; as a result, $\omega \to n{\omega _0}$, prefactor $g \to g/(2\pi {T_0})$, where the $1/(2\pi)$ comes from the 1D nature of our model. The harmonic yield becomes $|{h_n}{|^2} = |{\tilde j_{\textit{er}}}(n{\omega _0}{)|^2}$ with ${T_{\textit{jl}}}$ given by the WQC propagator [Eq. (12a)].

In Fig. 2, the blue empty circles (exact) and blue filled circles (WQC) refer to results for the weakly bound model semiconductor, with $\Omega = 0.5$, and ${\omega _0} = 0.01425$. Red empty squares (exact) and red filled squares (WQC) refer to the tightly bound semiconductor, with $\Omega = 1.5$, and ${\omega _0} = 0.0285$. Plots with the same symbols in Figs. 2(a) and 2(b) correspond to the same values of $\Omega$ and ${\omega _0}$, but differ in ${F_0}$.

The WQC results are only plotted above the minimum bandgap, as the quasi-classical analysis is limited to processes for which an electron is actually born in the conduction band. Below the minimum bandgap, virtual processes dominate; in this range, the intraband current can have a substantial contribution that is not shown here.

The WQC approach agrees well with the exact solution, with most data points being off by less than a factor of 2. Even the first 1–2 cutoff harmonics are described fairly well, which demonstrates that they are of quasi-classical origin. The good agreement allows us to interpret semiconductor quantum dynamics such as ionization, electron/hole transport, and HHG in terms of classical trajectories. The quantum contributions to HHG are captured by the tunneling exponent ${t_x}$, by the pre-exponential factor $g$ in Eq. (12a), and by the Wannier dipole moments in Eq. (7).

A few points disagree by a larger factor of up to 6. In particular, Fig. 2(a) shows that the WQC result for harmonic $n = 15$ exhibits larger discrepancy for the weakly bound semiconductor ($\Omega = 0.5$) compared to the more tightly bound semiconductor ($\Omega = 1.5$). The reason for this behavior is identified in Fig. 3 and will be discussed later.

 

Fig. 3. Contribution of the long and short classical trajectories to the probability amplitude $|{P_{\textit{jl}}}|$ for harmonic order $n = 15$ in a wideband semiconductor; parameters $a = 7$, $\Omega = 0.5$, $\omega = 0.01425$, and ${F_0} = 0.0025$ corresponding to filled blue circles in Fig. 2(a). (a) depicts the combinations of birth ($l$) and recombination ($j$) site indices for which each trajectory exists and contributes to ${P_{\textit{jl}}}$; black regions indicate no solution. (b) shows the contribution of the long trajectory to $|{P_{\textit{jl}}}|$, while (c) shows the contribution from the short trajectory. Note that the values of the color scale differ by 2 orders of magnitude in (b) and (c).

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Numerical solution of the full saddle-point equations reveals two distinct classical trajectories that contribute to the probability amplitude ${P_{\textit{jl}}}$: one long trajectory and one short. Moreover, each solution exists for only certain combinations of birth ($l$) and recombination ($j$) lattice sites. Figure 3 shows the contributions arising from the different classical trajectories for the 15th harmonic $(n = 15)$ with $\Omega = 0.5$, ${F_0} = 0.0025$, corresponding to the filled blue circles in Fig. 2(a). Figure 3(a) depicts the regions in the $j-l$ plane, where each trajectory contributes to $|{P_{\textit{jl}}}(n = 15)|$. No solution exists for the dark region in the top-right, and the probability amplitude here is zero. Figures 3(b) and 3(c) show the individual contributions to the probability amplitude from the long and short trajectories, respectively. The long trajectory is dominant, as the electron–hole pair is born close to the field peak, whereas the short trajectory is born closer to the nodal point. This outweighs the effect of the short dephasing time, which favors the short trajectory. As a result, the contribution of each data point to the WQC propagator is dominantly determined by a factor ${\sim}g{e^{- {t_x}}}$ of a single (long) trajectory. The full probability amplitude $|{P_{\textit{jl}}}(n = 15)|$ is essentially identical to that of Fig. 3(b).

In Fig. 4, the total probability amplitude for the 15th harmonic $|{P_{\textit{jl}}}(n = 15)|$ is plotted as a function of birth and recombination site indices $l,j$ for $\Omega = 1.5$, and ${F_0} = 0.008$, which corresponds to the filled red squares in Fig. 2(a). For this system, the long trajectory is also dominant, and analysis of the individual contributions would reveal a picture qualitatively similar to Fig. 3.

 

Fig. 4. Probability amplitude $|{P_{\textit{jl}}}|$ versus birth ($l$) and recombination ($j$) site indices for harmonic order $n = 15$ in a narrowband semiconductor; parameters $a = 7$, $\Omega = 1.5$, $\omega = 0.0285$, and ${F_0} = 0.008$ corresponding to filled red circles in Fig. 2(a). Here we plot the total probability amplitude $|{P_{\textit{jl}}}|$, but note that the long trajectory is dominant; the individual contributions are similar to the behavior depicted for the wideband semiconductor in Fig. 3(b).

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In both Figs. 3 and 4, harmonic $n = 15$ has been selected, as the WQC result for the weakly bound semiconductor exhibits a more pronounced difference, while it agrees well for the tightly bound semiconductor. For both systems, the maximum probability is shifted towards negative birth-site indices; it is more likely for electron and hole to be born apart than at the same site. Tunnel ionization probability is determined by ${e^{- {t_x}}}$ and by birth dipole moment $d_l^*$. The tunnel exponent ${t_x}$ depends on the ionization potential ${E_g} + F({t_b}){x_l}$; see Eq. (12a). Thus, for the positive field, the electron–hole pair gains energy when born at increasingly negative distances, which reduces ${t_x}$. When ${-}{x_l} = {E_g}/F({t_b})$, ${t_x}$ vanishes; in other words, the valence and conduction band levels separated by ${-}l$ sites align, and the electron hops from the valence to the conduction band site. The penalty to be paid is a rapidly dropping dipole moment ${d_l}$. As such, the birth-site index at which ionization is maximum is determined by a trade-off between tunnel exponent and Wannier dipole moment. The dipole elements for the parameters of Fig. 3(a) drop more slowly with increasing $|l|$ than for (b); see Appendix B. Therefore, the site of highest ionization probability is shifted more strongly towards negative $l$. Recombination is most probable for $j = 0$ in Figs. 3(a) and 3(b), which is consistent with previous findings [46]. The drop in probability for increasing $j$ is due to ${d_j}$, which is why $|{P_{\textit{jl}}}|$ extends to larger $j$ in Fig. 3(b).

The results in Figs. 3 and 4 are displayed for birth times in the positive field cycle $0 \le {t_b} \le {T_0}/2$; the negative half-cycle would show the same picture, but mirrored about the $x -$ and $y -$ axis ($j,l \to - j, - l$).

Recall that exact and quasi-classical results do not agree well for harmonic $n = 15$ in Fig. 2(a) ($\Omega = 0.5$). The reason is found in Fig. 3(b); disagreement is due to the point $(j,l) = (4, - 2)$, which exhibits unusually high probability. We find that at this point ${k_s}$ is approximately zero, and therewith $|{\cal H}| \approx 0$. Since $g \propto 1/\sqrt {|{\cal H}|}$, this leads to a large value of the prefactor $g$. This behavior indicates that the quadratic saddle-point expansion is no longer sufficient, and the next higher order term(s) must be included. The rules of saddle-point integration require expanding the exponent up to the first nonvanishing term. In contrast, agreement for harmonic $n = 15$ in Fig. 2(a) for $\Omega = 1.5$ is good. This is consistent with the fact that in Fig. 4, ${k_s} \approx 0$ does not occur in areas of high probability.

Finally, the WQC method hinges on saddle-point integration, which works well when the exponent is rapidly oscillating. This is fulfilled for wideband semiconductors with large bandwidth ($\Delta$) and in the long wavelength limit. When transitioning to smaller $\Delta$ (dielectrics) and shorter wavelengths, saddle-point integration is expected to fail at some point. This will be subject to further research. Also, it is generally possible for transitions involving higher conduction bands to contribute to the harmonic spectrum, but this is beyond the scope of the two-band model considered here.

4. CONCLUSION

In summary, we have shown that the full single-body quantum dynamics driving HHG in wideband materials, such as semiconductors, can be quantitatively explained in terms of quasi-classical trajectory propagation. The physical insight offered by trajectory analysis will prove useful for optimization and the design of strong field and attosecond experiments and for the development of novel diagnostic applications of HHG, such as reconstruction of the dipole moment [58]. We believe that our approach presents a versatile tool for investigating open issues in strong-field solid-state physics, such as the role of noise and many-body effects in strong-field processes. Beyond that, quantitatively accurate quasi-classical analysis should be of interest for a wider range of topics in material science.

While our work is limited here to centrosymmetric materials and two-band model solids, we believe that the WQC method can contribute to understanding HHG in more complex materials. Multiband dynamics can be incorporated following the approach outlined in Ref. [18]. Noncentrosymmetric materials exhibit an interband transition dipole phase as well as an intraband Berry phase, which lead to additional terms in the classical action. The anomalous velocity arising from the Berry phase will offset the electron–hole trajectory in the plane perpendicular to the laser polarization axis. The Bloch quasi-classical model requires the electron–hole pair to recombine at the same lattice site, thus missing important trajectories where it could recombine from nearby lattice sites.

APPENDIX A: HESSIAN

Here we provide expressions for the determinant of the Hessian ${{\cal H}_{\textit{ij}}} = {\partial ^2}\varphi /{\partial _i}{\partial _j}$ appearing in Eq. (12). Evaluation of the second derivatives yields

(A1)$$|{\cal H}| = \left| {\begin{array}{*{20}{c}}{{\bf F}(t^\prime) \cdot {\bf v}({\bf k}(t^\prime ,t))}&\quad{- {\bf F}(t) \cdot {\bf v}({\bf k}(t^\prime ,t))}&\quad{{v_x}({\bf k}(t^\prime ,t))}&\quad{{v_y}({\bf k}(t^\prime ,t))}&\quad{{v_z}({\bf k}(t^\prime ,t))}\\{+ \dot{\boldsymbol F}(t^\prime) \cdot {{\bf x}_l}}\\{- {\bf F}(t) \cdot {\bf v}({\bf k}(t^\prime ,t))}&\quad{{\bf F}(t) \cdot {\bf v}({\bf k}) - \dot{\boldsymbol F}(t) \cdot {{\bf x}_j} -}&\quad{- {v_x}({\bf k}) +}&\quad{- {v_y}({\bf k}) +}&\quad{- {v_z}({\bf k}) +}\\ &\quad{{F_i}(t){D_{\textit{ij}}}(t^\prime ,t){F_j}(t)}&\quad{{F_i}(t){D_{\textit{ix}}}(t^\prime ,t)}&\quad{{F_i}(t){D_{\textit{iy}}}(t^\prime ,t)}&\quad{{F_i}(t){D_{\textit{iz}}}(t^\prime ,t)}\\{{v_x}({\bf k}(t^\prime ,t))}&\quad{- {v_x}({\bf k}) + {F_i}(t){D_{\textit{xi}}}(t^\prime ,t)}&\quad{- {D_{\textit{xx}}}(t^\prime ,t)}&\quad{- {D_{\textit{xy}}}(t^\prime ,t)}&\quad{- {D_{\textit{xz}}}(t^\prime ,t)}\\{{v_y}({\bf k}(t^\prime ,t))}&\quad{- {v_y}({\bf k}) + {F_i}(t){D_{\textit{yi}}}(t^\prime ,t)}&\quad{- {D_{\textit{yx}}}(t^\prime ,t)}&\quad{- {D_{\textit{yy}}}(t^\prime ,t)}&\quad{- {D_{\textit{yz}}}(t^\prime ,t)}\\{{v_z}({\bf k}(t^\prime ,t))}&\quad{- {v_z}({\bf k}) + {F_i}(t){D_{\textit{zi}}}(t^\prime ,t)}&\quad{- {D_{\textit{zx}}}(t^\prime ,t)}&\quad{- {D_{\textit{zy}}}(t^\prime ,t)}&\quad{- {D_{\textit{zz}}}(t^\prime ,t)}\end{array}} \right|,$$

where the determinant is evaluated at the saddle point defined by $t^\prime = {t_b} + i\delta ,t = {t_r},{\bf k} = {{\bf k}_s}$. Using linear dependence between column 2 and columns 3, 4, and 5 (see the supplement of [18]), the determinant can be simplified to

(A2)$$|{\cal H}| = \left| {\begin{array}{*{20}{c}}{{\bf F}(t^\prime) \cdot {\bf v}({\bf k}(t^\prime ,t)) + \dot{\boldsymbol F}(t^\prime) \cdot {{\bf x}_l}}&\quad 0&\quad {{v_x}({\bf k}(t^\prime ,t))}&\quad {{v_y}({\bf k}(t^\prime ,t))}&\quad{{v_z}({\bf k}(t^\prime ,t))}\\{- {\bf F}(t) \cdot {\bf v}({\bf k}(t^\prime ,t))}&\quad{- \dot{\boldsymbol F}(t) \cdot {{\bf x}_j}}&\quad{- {v_x}({\bf k})\; +}&\quad{- {v_y}({\bf k})\; +}&\quad{- {v_z}({\bf k})\; +}\\{\,}&\quad{\,}&\quad{{F_i}(t){D_{\textit{ix}}}(t^\prime ,t)}&\quad{{F_i}(t){D_{\textit{iy}}}(t^\prime ,t)}&\quad{{F_i}(t){D_{\textit{iz}}}(t^\prime ,t)}\\{{v_x}({\bf k}(t^\prime ,t))}&\quad{- {v_x}({\bf k})}&\quad{- {D_{\textit{xx}}}(t^\prime ,t)}&\quad{- {D_{\textit{xy}}}(t^\prime ,t)}&\quad{- {D_{\textit{xz}}}(t^\prime ,t)}\\{{v_y}({\bf k}(t^\prime ,t))}&\quad{- {v_y}({\bf k})}&\quad{- {D_{\textit{yx}}}(t^\prime ,t)}&\quad{- {D_{\textit{yy}}}(t^\prime ,t)}&\quad{- {D_{\textit{yz}}}(t^\prime ,t)}\\{{v_z}({\bf k}(t^\prime ,t))}&\quad{- {v_z}({\bf k})}&\quad{- {D_{\textit{zx}}}(t^\prime ,t)}&\quad{- {D_{\textit{zy}}}(t^\prime ,t)}&\quad{- {D_{\textit{zz}}}(t^\prime ,t)}\end{array}} \right|.$$

Here, $i,j \in \{x,y,z\}$, summation is implied when indices $i$ or $j$ are repeated, ${D_{\textit{ij}}} = \int_{{t^\prime}}^t \text{d}\tau {\beta _{\textit{ij}}}({\bf k}(t^{\prime \prime} ,t))$, ${\beta _{\textit{ij}}} = {\partial _{{k_i}}}{v_j}({\bf k})$, and $\dot{\boldsymbol F}(t) = {\partial _t}{\bf F}(t)$. For completeness, $|{\cal H}|$ is given for a general field ${\bf F}(t)$; for the case treated here, set ${F_y} = {F_z} = 0$. To leading order $|{\cal H}| = {v_x}({\bf k}){\bf f}(t^\prime ,t,{\bf k}) + \dot{\boldsymbol F}(t) \cdot {{\bf x}_l}h(t^\prime ,t,{\bf k})$, where $h,{\bf f}$ are minors of $|{\cal H}|$. For completeness, we have included time derivatives of the laser field, which are, however, small in the long wavelength limit. As a result, the leading order term is $|{\cal H}| = {v_x}({\bf k}){\bf f}(t^\prime ,t,{\bf k})$.

APPENDIX B: DELTA FUNCTION POTENTIAL

The WQC approach and its physical significance are explored by means of a 1D delta-function model potential, $V(x) = \Omega \sum\nolimits_{n = - \infty}^\infty \delta [x - (n + 1/2)a]$ with unit cell size $a$ and barrier penetration parameter $\Omega$. For the investigated parameters, the bandgap is well approximated by the nearest neighbor approximation, $\varepsilon = {E_g} + \Delta [1 - \cos (ka)]$, where ${E_g}$ is the minimum bandgap and $2\Delta$ represents the bandwidth.

The binding energy is determined by $2{E_m} = K_m^2$, where $m = v,c$ and ${K_m}$ is determined by

(B1)$$\cos (ka) = \cos ({K_m}a) + \frac{\Omega}{{{K_m}}}\sin ({K_m}a).$$

The wave function is given by

(B2)$$\begin{split}{\Phi _{m,k}}(x) &= \sqrt {\frac{1}{a}} {u_{m,k}}(x)\exp (ikx) \\ {u_{m,k}}(x)& = {A_m}(k)\left[{{e^{i({K_m} - k)x}} + {r_m}{e^{- i({K_m} + k)x}}} \right], \\ {A_m}(k)& = 1/\sqrt {1 + r_m^2 + 2{r_m}\sin ({K_m}a)/({K_m}a)} , \\ {r_m}(k) & = \frac{{\sin [({K_m} - k)a/2]}}{{\sin [({K_m} + k)a/2]}}.\end{split}$$

 

Fig. 5. (a) Bloch dipole transition elements $\text{Im}[{d^ *}(k)]$ versus $k$; (b) Wannier dipole transition elements ${d_j}$ versus $j$, which represents the difference in lattice sites at which electron and hole are born. 1D model parameters, $a = 7$; $\Omega = 0.5$ (black); $\Omega = 1.5$ (red).

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From the wave function, the Bloch dipole moment is found to be

(B3)$$\begin{split}{d_{\textit{cv}}}(k)& = {d^*}(k) = i\frac{{2{A_c}{A_v}}}{{{E_c} - {E_v}}} \\ &\quad\times \left\{{[({K_v} - k){r_c} - ({K_v} + k){r_v}]\frac{{\sin [({K_v} + {K_c})a/2]}}{{({K_v} + {K_c})a}} } \right. \\ &\quad + \left. {[({K_v} - k) - ({K_v} + k){r_v}{r_c}]\frac{{\sin [({K_v} - {K_c})a/2]}}{{({K_v} - {K_c})a}}} \right\}\text{.}\end{split}$$

We chose $a = 7$ and $\Omega = 0.5,1.5$ to model a weakly and more tightly bound semiconductor, respectively. The corresponding bandgap parameters are ${E_g} = 0.141,0.269$ and $\Delta = 0.269,0.17$. The Bloch dipole elements $d(k)$ and Wannier dipole elements ${d_j}$ are plotted in Fig. 5. As expected, ${d_j}$ drops faster for the more tightly bound model. Finally, we have chosen the coordinate center at the point of inversion symmetry that corresponds with choosing a maximally localized Wannier basis [56]. For this choice, the diagonal (intraband) dipole moments are zero, and the phase of the interband dipole moment is constant.

Funding

Air Force Office of Scientific Research (FA9550-16-1-0109).

Acknowledgment

G. Ernotte was supported by the Vanier Canada Graduate Scholarship program. T. Brabec acknowledges the support of the NSERC DG and CRC programs.

Disclosures

The authors declare no conflicts of interest.

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